Math Knowledge - Third Attempt
Volume of a triangular prism
(2) b = 1/2bh (1) v = Bh
Volume of a Pyramids
(2) v=1/3Bh (1) a = lw
Area of the Trapezoids: Base One: 2 ft Base Two: 4.5 ft Height: 2 ft
1/2 (2 ft + 4.5 ft) 2 ft = 6.5 ft
How many different license plates can be created that contain three letters and three numbers?
26 * 26 * 26 * 10 * 10 * 10 = 17, 576, 000
Factoring Completely
8x^2 - 24x + 16 From now on, for any factoring problem you do, your first step will be to determine if a GCF exists for the original polynomial.
Surface area of the following prism
A = 2(wh + lw + lh) Area of top & bottom Area of left & right Area of front & back
Surface area of the following pyramid.
A = s^2 A = 1/2(b)(h) A = 4(1/2(b)(h) SA = 4(1/2(b)(h) + s^2
Monomial
A polynomial with just one term. 3xy^2
Proportion
An equation stating that two ratios are equal a) 5/4 = x/12 (5)(12) = (4)(x) b) x-5/x+4 = 2/3 3(x - 5) = 2(x+4)
Probability of Dependent Events
Changes depending on the outcomes of other events Ex1: P(brown, then brown) P(brown) = 4/12 = 1/3 P(brown) = 3/11 (1/3)(3/11) = 3/11 = 1/11
Solving Proportions
Cross multiply, then divide Solve the following proportion. 9/15 = 6/d (9)(d) = (15)(6) 9d = 90 d = 10
Writing Radicals in Exponential Form
Ex1: (^11√2)^13 2^13/11 *Note: However you change the exponential form to radical is the same form by reverting back by getting the same given form. Ex2: (^7√6)^4 Answer: 6^4/7 Ex3: ^9√11^12 11^12/9 Answer: 11^4/3 *Note: The root of the radical is the denominator of the exponent, and the power is the numerator.
Angle-Angle Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. (AA~)
Area of a Squares and Rectangles
The area of a figure is the amount of surface that is covered by the figure. Area = Length x Width A = lw A = s^2
Simplify the expression: 3x + 6.5x^2 = ?
This is expression cannot be simplified.
Pythagorean Theorem
a²+b²=c² The Pythagorean Theorem: The sum of the squares of the lengths of the legs of a right triangle is equal to the square of the length of the hypotenuse.
Rachel is driving to visit her mother, who lives 250 miles away. How long will the drive be, round trip, if Rachel drives at an average sped of 40 mph?
d = rt 500 = 40t 12.5 = t If takes Rachel 12.5 hours to make the round-trip drive. Multiply 60 to find the answer in minutes. 12.5(60) = 750 minutes.
Distance Formula
d = √[( x₂ - x₁)² + (y₂ - y₁)²] Note: Don't get confused (x,y) (x,y) (1,1) (2,2)
Volume of Cones
v=1/3π(r^2)h
Volume of Spheres
v=4/3πr³
Evaluating and Graphing Functions
In the equation y = x + 1, the value of y depends on the value of x. The value of y is a function of the value of x. y = x + 1 f(x) = x + 1 Extra note: Create a t-chart for the following function, and graph the function. y = x + 2
Wing and Current Word Problems
R * T = D p + w (tailwind) p - w (headwind)
Motion Word Problems
Rate * Time = Distance Problem 1: Two cars leave from the same place at the same time and travel in opposite directions. If one is travelling at 60 mph and the other at 65 mph, after how many hours will they be 625 miles apart? D1 + D2 = 625 Problem 2: Two bikers leave from cities that are 230 miles apart and travel toward each other. 5 hours later, they meet. If one is travelling 10 mph faster than the other, how fast is each travelling? D1 + D2 = 230 Problem 3: A blue truck going 50 mph is follow 3 hours later by a red truck going 60 mph. After how many hours will the red truck overtake the blue truck? D(blue) = D(red) Note: "Later" mean (minus/subtract)
Rational/Irrational
i) 0 ii) 0.515115111 i) All integers are rational. ii) When a decimal is both non-terminating and non-reapeating, it is irrational.
Volume of a cube
v=lwh
Volume of Cylinders
v=π(r^2)h
Metric Unit Conversions
1 Kilometer 1 Hectometer 1 Dekameter 10 Decimeters 100 Centimeters 1,000 Millimeters King Henry Died By Drinking Chocolate Milk Kilo means 1,000 Hecto means 100 Deka means 10 Deci means 1/10 Centi means 1/100 Milli means 1/1,000 My Large Giraffe Meter: Distance Liter: Capacity Gram: Weight
If the mean of the following data set is 4, find the missing number in the data set. 3, 0, 4, 14, 1, _
3, 0, 4, 14, 1, x 4 = 3+0+4+14+1+x /6 2 = x
Baskin Robbins sells 31 flavors of ice cream. How many ways can a customer order two scoops of ice cream?
31 * 31 = 961
The range of 7 numbers is 8. The greatest number is 99, the meidian is 94, and the mode is 96. What are the numbers? (Hint: All numbers in this problem must be integers.)
91, 92, 93, 94, 96, 96, 99 *Note that this is the only possible answer.
Customary Unit Conversions
Time, Length, Weight, Liquid Volume. Larger to Smaller (multiply) Smaller to Larger (divide)
Probability of Independent Events
(Indepedent events) A coin is tossed twice. Find the probability of the following event: P(A and B) = P(A) * P(B) Ex1: P(heads and heads) P(heads) * P(heads) 1/2 *1/2 = 1/4 Ex2: P(red, then green) P(red) = 3/6 = 1/2 P(green) = 1/6 (1/2)(1/6) = 1/12 Ex3: P(green or yellow, then yellow) P(green or yellow) * P(yellow) 5 green + 4 yellow = 9/20 4 yellow = 4/20 *Remember 1 is not a prime number.
Similar Polygons
(polygons that have the same shape) (~) = Similar sign (≅) = Congruent If two polygons are ~, then corresponding ∠'s are ≅, ... and lengths of corresponding sides are in proportion. The scale factor is the ratio of the lengths of two corresponding sides. AE/RV = 18/12 = 3/2
Jared orders a box of dozen doughnuts, and each doughnut is a different type. How many different ways can the doughnuts be arranged?
12*11*10*9*8*7*6*5*4*3*2*1 479,001,600 ways Remember that a "dozen" means 12.
Perimeter and Circumference Word Problems
A baseball diamond is a square with sides of length 58 ft. What is the perimeter of the baseball diamond? Answer: 232 ft What happens to the perimeter of a rectangle if you double the length of each side? Answer: Perimeter double A regulation NBA basketball hoop has a circumference of 53.38 inches. What is the diameter of the hoop? Answer: C = πd What happens to the circumference of a circle if you double the length of it diamter? Answer: Circumference doubles
Circle Vocabulary
A circle is a set of points in a plane that are a given distance from a given point in the plane. All radii of a given circle are congruent. A radius is a segment that joins the center of the circle to a point on the circle. A chord is a segment whose endpoints lie on the circle. A diameter is a chord that contains the center of the circle. A secant is a line that contains a chord, or a line that intersects the circle at two points. A tangent is a line in the plane of a circle that intersects the circle at one point.
Polygon
A flat shape with many straight sides. Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up).
Vowels
A, e , i, o, u and sometimes y and sometimes w
Area of a Triangle
A= 1/2 bh Perpendicular: At right angles (90*) to. *Note: The base of the triangle is the side perpendicular to the height.
Area of the Trapezoids
A=1/2(b1+b2)h A flat shape with 4 straight sides that has a pair of opposite sides parallel.
Area of Parallelograms
A=bh A flat shape with 4 straight sides where opposite sides are parallel.
Area of a Circle
A=πr²
Consonant
B, C, D, F, G, H, J, K, L, M, N, P, Q, R, S, T, V, X, Z, and usually W and Y.
Congruent Arces
Congruent arcs are arcs that have the same measure (AB = CD). In the same circle or congruent circles, if two central angles are congruent, then their minor arcs are congruent. If ∠ AOB = ∠ COD, then AB = CD. In the same circle or congruent circles, if two minor arcs congruent, then their central angles are congruent.
Arcs and Central Angles
Central Angle, Arc, Minor Arc, & Major Arc ∠ AOB is a central angle of ⦿ O. - A central angle is an angle that has a vertex at the center of circle. AB and AXB are arcs of ⦿ O. An arc is composed of two points on a circle (the endpoints) and the points on the circle that connect the two endpoints. AB is a minor arc of ⦿ O. A minor arc is composed of two endpoints and the points on the circle that are on the interior of its central angle. The measure of a minor arc is equal to the measure of its central angle (mAB = 70*). AXB is a major arc of ⦿ O. A major arc is composed of two endpoints and the points on the circle that are not on the circle that are not on the interior of its central angle. The measure of a major arc is equal to 360 minus the measure of its central angle (mAXB = 360 - 70 = 290*)
Units of Measurement
Converting between Customary Units and Metric Units Length: 1 mile = 1.6 kilometers 1 inch = 2.54 Weight: 1 ounce = 28.4 grams Volume: 1 liter = 2.1 pints Convert the following 25 mi = _m 1 mile = 1.6 kilometers 25 mi * 1.6 = 40 km 1 kilometer = 1,000 meters 40 km * 1,000 = 40,000 m
Corresponding
Equivalent in character, form, or function; comparable.
Multiplying and Dividing with Scienctific Notation
Ex1: (1.4 x 10^-2)(5.3 x 10^6) [7.42 x 10^4] Note: You need to times the decimals together and than times the 10 and add the exponent because the power of rule is that you add the exponent if it's timing. Ex2: 6.5 x 10^3 ----------- 9.8 x 10^9 0.663265306 x 10^-6 = 0.66 x 10^-6 6.6 * 10^-7 Note: You need to divid the number and than subtract the exponent as given. Once you have your answer for that you'll need to times 10 on the decimal number because that number need to always be a whole number from 1 to 9 and if you times the decimal, you'll need to divid the number 10 which mean the denominator number has an exponent of 1 and the power of rule is if both exponent are being divid, that need to be subtract. Ex 3: (3.9 x 10^5)^2 = (3.9 x 10^5)(3.9 x 10^5) = 15.21 x 10^10 = 1.521 x 10^11 Note: If you ever get a decimals more than 9, all you need to to is divide it by 10 and then times the number 10 and the power of rule say that you'll add whatever the number you are given.
Numerical Bases with Rational Exponents
Ex1: 3^1/3 * 9^1/3 3^1/3 * (3^2)^1/3 3^1/3 * 3^2/3 3^1/3 + ^2/3 3^3/3 3^1 3 *Note: we need to have like bases to add the exponents so we simply just sqaure the base number. And leaving the fraction exponent outside of the parentheses. After that you multiply the base exponent and fraction exponent. When you get rid of the parentheses, you will then add the both exponent on the base. Ex2: 4^3/4 ------- 4^1/4 Answer: 2 *Note:
Rational Exponents
Ex1: 81^1/2 An exponent of 1/2 means that we take the square root of the base. √81 = 9 Ex2: 16^3/4 = (^4√16)^3 = (2)^3 = [8] Note: The denominator of the exponent gives us the root, and the numerator of the exponent gives us the power. The denominator of the exponent gives us the root, and the numerator of the exponent gives us the power. Ex3: - 3^√64 = [-4] We take the cube root of 64, not the cube root of -64, because the -64 is not in parentheses.
Equation of a Circle
Ex1: Center (-2, 7), radius 5 Center (-2, 7) = (h, k) Radius (5) = (r) (x-h)²+(y-k)²=r² [x-(-2)]² + [y-(7)]² = (5)² Answer: (x+2)² + (y-7)² = 25 Ex2: Find the center and the radius of the following circle, then graph the circle. (x-4)² + (y+3)² = 64 (x-h)² + (y-k)² = r² Note: Match and solve h = 4, k= -3 r² = √64 = 8 Center: (4, -3) Radius: (8)
Premutations
Find the number of permutations of the letters A, B, and C. ABC, BAC, CAB ACB, BCA, CBA 6 Permutations or 3 (letters) * 2(cannot use same letter) * 1(lead to one letter only) = 6 Ex2: How many ways can 5 people stand in a line? 5*4*3*2*1 = 120 (Is an arragement of obejects in order of important)
Surface area of a following cube
Find the sum of all the faces. Find the area of each face A = s^2 SA = Side * Area given
If a square is cut in half across its diagonal, the formed triangles will have angles of:
If a square is split in two at its diagonal, then its angles will also be bisected. A bisection of a 90 degree angle is 45 degrees. Thus, the formed triangles would have angles of 45, 45, and 90 degrees. *Note: 90/2 = 45 degrees - Sqaure angles add up to 360 degree.
At the arcade, Nick spent 203 tokens on videogames in seven days. Each day he spent five more tokens than the previous day. How many tokens did Nick spend on the third day at the arcade?
Let x = the number of tokens Nick spent on day 1. Set up the following equation and solve for x: x + (x + 5) + (x + 10) + (x + 15) + (x + 20) + (x + 25) + (x + 30) = 203 7x + 105 = 203 x = 14 tokens We know Nick spent 14 tokens on the first day. To find what he spent on the third day, add 10 tokens to get 24 tokens.
Combinations
List all of the combinations that can be formed by choosing two of the following letters: A, B, C, and D. AB AC AD BC BD CD (A combination is an arrangement of objects in which order is not important.)
Median
The middle number in the data set when data is written from least to greatest.
A combination lock has a 3-digit combination made from the digits 0-9 without using the same digit twice. How many arrangements are possible? Decide whether this is a permutation or combination.
Permutation 10 * 9 * 8 = 720
Midpoint Formula
The midpoint of the line segment joining any two points (x₁+x₂) and (y₁+y₂) is: (x₁+x₂)/2, (y₁+y₂)/2 Note: (x, y) (x, y) (1, 1) (2, 2)
Semicircle
Semicircle: If AB is a diameter of the circle, then AXB is a semicircle. A semicircle is an arc whose endpoints are the endpoints of a diameter of the circle. A semicircle can also be thought of as a circle, so the measure of a semi is 1/2 * 360, or 180* AB and BC are adjacent arcs. Adjacent arcs of a circle are arcs that intersect at exactly one point. The Arc Addition Postulate states that mAB + mBC = mAC mAC = 80 + 30 = 110
Mike made a snack mix by combining 1 pound of nuts that cost $13.50 per pound; 2 pounds of pretzels that cost $4.25 per pound; and 4 pounds of cheese crackers that cost $4.60 per pound. What is the price per pound of Mike's snack mix?
Since all of the ingredients are in $ per pound, we just need to do some multiplying and adding. 1 pound nuts * $13.50 per pound = $13.50 2 pounds of pretzels* $4.25 per pound = $8.50 4 pound of cheese crackers * $4.60 per pound = $18.40 Add the pounds 1 (nuts) + 2 (pretzels) + 4 (crackers) = 7 pounds of mix were made. Add the cost of each: $13.50 + $8.50 + $18.40 = $40.40 is the cost for all 7 pounds of ingredients. If you need the price per pound, take our cost for the whole mix ($40.40) and divide it by the number of pounds (7) and you will have the price per pound. $40.40/7 = $5.77 per pound.
Tree Diagrams and Counting Principle
The Counting Principle states that if there are "a" ways for one event to occur and "b" ways for a second event to occur, then there are "a * b" ways for both events to occur.
Jake's final grade in math class is determined by the average he gets on six tests in the class. So far, Jake has earned a 97, 89, 85, 90, and 99 on the first five tests. What grade must Jake earn on the sixth test in order to get a 93 average in the class?
The average is found by dividing the total of all scores and dividing it by the number of scores. Since we know the average Jake wants (93), and we have all the scores from Jake's tests except one, we can set up an equation, with x representing Jake's score on the 6th test. 97 + 89 + 85 + 90 + 99 + x = 93 6 460 + x = 558 x = 98
Charles walked all the way around a crop circle and found that traveled 5.5 miles. If he had walked directly through it and passed the center, how much shorter would he have walked?
The circumference = diameter x pi and The diameter = circumference/pi d = 5.5/3.14 The diameter is found to be 1.75 miles, meaning if he had walked straight through, He would have walked 5.5 - 1.75, or 3.75 miles shorter.
Range
The difference between the largest number and the smallest number in the data set.
Circumference
The distance around a circle Circumference = 2 * π * radius C = 2πr π = 3.14 or 22/7
Mean
The mean is the average of the numbers: a calculated "central" value of a set of numbers. To calculate it: add up all the numbers, then divide by how many numbers there are.
Mode
The number in the data set that appears most often.
Multiplying and Dividing with Nagative Exponents
The power rule is that when dividing, your exponents get subtracted and when your multiplying your exponents add together and when you have a parenthesis with an exponent outside you would multiply so don't get confused my good friend. And if there is a negative exponent than you need to bring it down to the denominator and switch it to a positve and add a one on the numerator and if the base number has a zero then you would just change that number to a one. Note: don't get confused with parenthesis times parenthesis, if both bases are the same, you'll just add it. Same concept apply with the power of rule with multiplying my friend. Don' think too hard. just keep it simple. Note: when you get a negative exponent on the denominator you can bring it to the numerator to be a positive. And vice versa with a negative exponent on the numerator, you can bring it to the numerator to change it to a postive my friend. Note: my friend when there is a parenthesis around a fraction and an exponent externally, you need to times all the bases, don't focus on a letter attaching to a number.
Perimeter
The sum of the lengths of the sides of a polygon
Volume of a Prisms
V=lwh
Numerical Bases with Negative Exponents
Write the following in simplest form without negative or zero exponents. Ex 1: 6^-7 * 6^5 = 6^-7 + 5 = 6^-2 = 1/6^2 = 1/36 Ex 2: 5/3^-1 = 5*3^1 = 5*3 = 15 The rule is that when there's a negative eponent on the denominator, you bring that to the numerator to make it a positive and solve the number as usual to get you a answer without the exponent. And then you time the two number. Ex 3: (-2^-1)^3 = -2^-1 * 3 = -2^-3 = 1/-2^3 = 1/-8 The rule is that you multiply the power outside the bracket into the exponent. and if you get a negative exponent, you can swap it to a positive and put a "1" as the numerator while the primary or base number can stay negative or positive depending what problem is given. Ex 4: (1/7)^-2 = 1^-2 / 7^-2 = 7^2/1^2 = 49/1 = 49 The rule is that when you have the both exponent on the numerator and denominator you can swap both side and change it as a positive and solve the answer from there. Ex 5: (1/2)^-3 = 1^-3 / 2^-3 = 2^3 / 1^3 = 8 The rule is that if there are two exponents on the fraction than you can swap the two primary number and turn it into positive. Ex 6: (- 3/8)^2 = (-3)^-2 / (8)^-2 = (8)^2 / (-3)^2 = (8)(8)/(-3)(-3) = 64/9 The rule is that parathesis is important aspect because you can use negative however the exponent is given, but if there is no exponent than you do not need to worry.
In each of the following problems, determine whether you would need to find the number of permutations, combinations, or simply apply the counting principle.
a) George is trying to decide how to spend the day in Istanbul. In the morning, his options are the Grand Bazaar, Topkapi Palace, or Hagia Sofia. In the afternoon, his options are a ferry ride, a Turkish Bath, or a tea house. How many different ways can George spend the day in Istanbul? Answer: Counting Principle 3 * 3 = 9 b) How many ways can you arrange the letters in the word ISTANBUL? Answer: Permutations 8*7*6*5*4*3*2*1 = 40,320 c) How many different groups of two letters can be chosen from the word ISTANBUL? Answer: Combinations 28
If there are 11 numbers in a data set, and the value of the smallest number is decreased, determine which of the following values will change.
a) Mean = sum numbers/11 - The numerator of the mean will decrease, while the denominator will stay the same. [Decrease] b) Median > middle number - If we decrease the smallest number in a data set, this has no impact on the middle number in the data set. [No change] c) Mode > most common number - If smallest number is mode, then mode might change. - If smallest number is not mode, then mode won't change. [Cannot be determined] d) Range > largest minus smallest - If the smallest number is decreased, this will affect the range.
Determine whether each of the following events is dependent or independent.
a) Rolling a number cube twice. [Independent] b) Selecting 8 cards from a deck without replacing them. [Dependent] c) Two captains picking players for a basketball game. [Dependent] d) Taking a marble out of a bag, replacing it, mixing up the marbles, then taking another one out. [Independent]
Negative Exponents
ex 1. x^-7 * x^2 = x^-5 = 1/x^5 For multiplication, you need to add both number, depending on the positive and negative of it number To swap a negative number, input 1 over the number to make it a positive number. ex 2. (y^3)-3 = y^-9 = 1/y^9 For this case, you need to time both the number and when you get the number you need to put a number above the answer and change it to a positive. ex 3. b^8/b^-4 = b^8 - -4 = b^12 For this one, you'll need to add both exponent given. ex 4. (p^-2 q^11)^-1 = p^-2 * -1 q^11 * -1 = p^2 * q^-11 = p^2 / q^11 For this one take the power outside of the bracket and times it by the exponent given inside and from there you need to swap the negative number to a positive and you should get your number. ex 5. r^-3 s^-6 / r s = r^-4 s^-7 = 1/ r^4 s^7 You need to subtract the exponent on the numerator and denominator to get the answer and once you get the answer you need to swap the negative number to a positive number. ex 6. w^-9 * w * w^8 = w^-9 + 1 + 8 = w = 1 You need to add all the exponents to get the answer and once you get the answer such as "w" you can transition it to number 1. ex 7. (m^-1 / n^-5)^2 = m^-1 * 2 / n^-5 * 2 = m^-2 / n^-10 = n^10 / m^2 Same concept, just times the power outside of the bracket inside the exponent given and swap the negative to a positive. *Note: for this one you need to swap the denominator with the numeritor to be a positve demostrated as give at the top. ^Rule: if there is a negative on the opposite, just swap them. ex 8. (x/ y^-2 z^0)^-6 = (x)^-6/ (y^-2 z^0)^-6 = x^-6/y^12 z^0 = 1 / x^6 y^12 (1) = 1 / x^6 y^12 *Note on this one, the rule is that if there is a negative on the numerator you need to bring it to the denominator to make it a positve and a 1 on the numerator. ex 9. (p^-2/q)^-3 = p^6q^3 If the numerator exponent is positive than don't worry about it and if the denominator exponent is a negative than bring it up to the numerator and change it to a positive and combine with the current primary number.
Simple Probability
p(event) = no of favorable outcomes/ total no. of outcomes p(3) = 1/4, .025, 25%
Terri has a jar of orange, blue, green, and red chocolate candies. In the jar are 75 orange candies, 62 blue candies, and 52 green candies. If there is a total of 252 candies in the jar, what is the probability that Terri will pick a red candy?
¼ To solve this problem, first find the number of red candies in the jar by adding the numbers of all the other candies together, then subtracting from the total. Since there are 63 red candies out of a total of 250 candies, divide 63 by 252. Reduce the fraction by dividing the numerator and denominator by 63. 63/252 ÷ 63/63 = ¼
Coefficient
A number multiplied by a variable in an algebraic expression. A number used to multiply a variable. Example: 6z means 6 times z, and "z" is a variable, so 6 is a coefficient.
Absolute Value
How far a number is from zero. Example "6" is 6 away from zero, but "−6" is also 6 away from zero. The distance a number is from zero on a number line. ALWAYS POSITIVE result.
Given Lines are Parallel
If 2 parallel lines are cut by a transversal, then corresponding angles are congruent. If 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. If line l || m (l parallel m), then ∠ 1 = ∠ 2. Postulate: suggest or assume the existence, fact, or truth of (something) as a basis for reasoning, discussion, or belief. Theorem: A result that has been proved to be true (using operations and facts that were already known). If 2 parallel lines are cut by a transversal, then same-side interior angles are supplementary
Tara earns twice as much per hour as Kayte. Kayte earns $3 more per hour than Austin. As a group, they earn $41 per hour. What is Austin's hourly wage?
$8.00 If Austin's hourly wage is x, then Kayte's wage is (x+3), so Tara's wage is 2(x+3). Together they earn $41 per hour, so we can write the equation: x+(x+3)+2(x+3)=41 x+x+3+2x+6=41 4x+9=41 4x=32 x=8 Austin's hourly wage is $8 per hour
Advanced Multiplying Polynomials
(5x-2)(x^2-5x+3) = 5x^3 - 27x^2 + 25x - 6 (y-2)Y3 (y-2)(y-2)(y-2) (y^2 - 2y - 2y + 4)(y-2) (y^2 - 4y + 4)(y-2) y^3 - 2y^2 - 4y^2 + 8y + 4y - 8 y^3 - 6y^2 + 12y - 8
Multiplying and Dividing Rational Expressions
(ex1) (1) 5x^5 * x^2 - x - 20 _____________________________ x^2 + 7x + 12 * 10x^3 (2) 1x^2 * (x - 5) ______________________ (x + 3) * (2) (3) x^2(x-5) ___________ 2(x+3) (ex2) (1) 6 / 8 _____________ 7 / 21 (2) 9/4 (ex3) (1) x^2 - 15x + 54 * x^2 + 11x +28 ______________________________________ x^2 - 5x - 36 * x^2 + x - 42 (2) (x-9)(x-6) * (x+7)(x+4) _____________________________ (x-9)(x+4) * (x+7)(x-6) (3) 1 Note: If everything cancels out, the answer is 1.
What is the area of an isosceles right triangle that has one leg that measures 4 cm?
8 cm *Note: isosceles triangle has 2 sides equal; 2 congruent angles And it's asking for a right triangle with 2 sides and 2 congruent angles so the area of the triangle is: 1/2 (b)(h) 1/2 (4)(4) 1/2 (16) Answer: 8 cm
Simplifying Rational Expressions
(ex1) (1) x^2 + 9x + 18 __________________ x^2 + 4x - 12 (2) (x + 6)(x+3) ________________ (x + 6)(x - 2) (3) X+3 ______ x-2 (ex2) (1) a^2 - 5a ____________ a^2 - 25 (2) a(a-5) ________ (a+5)(a-5) (3) a ___ a+5 (ex3) (1) m^2 + 7m - 30 ____________________ m^2 - 3m (2) (m + 10)(m - 3) ____________________ m(m - 3) (3) m + 10 ________ m (ex4) (1) x^2 - x __________ x^3 - x (2) x(x-1) ____________ x(x^2 - 1) (3) x(x-1) _________ x(x+1)(x-1) (4) 1 _____ x+1
Beginning Polynomial Equations
(ex1) (x+5)(x-3) = 0 x + 5 = 0 or x - 3 = 0 x = -5 or x = 3 {-5, 3} (ex2) x^2 - 12x - 28 = 0 (x-14)(x+2) = 0 x-14 = 0 or x+2 = 0 x = 14 or x = -2 {14, -2} (ex3) 3n^2 + 17n = -20 3n^2 + 17n + 20 = 0 (3n + 5)(n + 4) = 0 3n + 5 = 0 or n + 4 n = -5/3 or n = -4 {-5/3, -4} (ex4) d^2 = 36 d^2 - 36 = 0 (d+6)(d-6) = 0 d + 6 = 0 or d - 6 = 0 {-6, 6} (ex5) 13s + 24 = 7s^2 0 = 7s^2 - 13s - 24 0 = (7s + 8)(s - 3) 7s + 8 = 0 or s - 3 = 0 s = - 8/7 or s = 3 {- 8/7, 3} Note: Always move terms to the side of the equation that will make the squared term positive.
Solving Systems by Addition
(ex1) 3x+y=5 7x-y=15 Answer: (2,-1) (ex2) 2a+5b=11 a-4b=-14 Answer: (-2,3) (ex3) 4c+3d=-9 7c+2d=-6 Answer: (0,-3) (ex4) -2x+3y=4 6x-9y=1 Answer: (0=13, which equal no solution 0 [null set]) 2x-y=4 -4x+2y=-8 Answer: 0 = 0 {(x, y): 2x-y=4}
Slope-Intercept Form
(ex1) Step One: y = 3/4x - 2 Step Two: y = mx + b Step Three: m=3/4, b=-2 Step Four: start plotting the y intercept (b-2) to plot on the y-axis and from there use the rise/run begining at point A to find point B. (ex2) Step One: y = -3x Step Two: y = mx + b Step Three: m = -3 Step Four: y = -3x + 0 Step Five: b = 0 Step Six: m = -3/1 Step Seven: m = -3/1, b = 0 (ex3) Step One: y = 4 Step Two: y = mx +b Step Three: y = 0x + 4 Step Four: m = 0, b = 4 Note: It may help, y = #, your graph will always be horizontal line. (ex4) y = -x - 1 Answer: m = -1/1, b = -1 (ex5) y = x Answer: m = 1/1, b = 0
Factoring Trinomials with Lead Coefficients and Positive Constants1
(ex1) 3j^2 - 16j + 21 Answer: (3j - 7)(j - 3)
A shopkeeper buys 12 boxes of apples for $90. If two boxes were lost, what price would the shopkeeper need to sell each remaining box of apples for to make a 20% profit?
$10.80 The shop keeper has lost 2 of his original 12 boxes. Therefore, the shopkeeper has 10 boxes left. Each box cost an average of: $90/10 = $9. If the shopkeeper wants a 20% profit he would need to sell each box at: $9 + 20% = $9 + ($9 x 0.2) = $9.00 + $1.80 = $10.80
Trinomial
A polynomial with three terms 3x^2 − 3y + 2
Addition and Subtraction Word Problems
Key Words: Addition> altogether, total, in all Total cost How many people altogether? Subtraction> left, remaining, more, fewer, increase, go up, grow, decrease, go down, reduce How much time is remaining? How many more does Sue have than Joe? Estimation> estimate, approximate
Multiplication and Division Word Problems
Key Words: Multiplication> times as many, times as much, find all... if each, in all, of Chris has $5 and John has 3 times as much Division> each per Cost per ounce Estimation> estimate, approximate
Problems Involving Perpendicular Lines
Lines that are at right angles (90°) to each other.
Factors and Primes
List all of the factors of the following number and state whether the number is prime or composite: 1 1/1=1 Neither Prime nor Composite
Gallon Word Problem
Stephen signed up to bring 5 gallons of lemonade to the company picnic. He has a 5-gallon bucket which contains 3.5 gallons of lemonade. How many pints of lemonade will he need to add in order to fill the bucket? He needs 1.5 gallons. There are 4 quarts in a gallon, so he will need 6 quarts. There are 2 pints in a quart, so he needs to add 12 pints.
Solving Systems by Substitution
Substitute, Solve and Substitute y = 3x+2 7x-4y = 7 Answer: (-3,-7) 2x+5b = -4 4b-a = -11 Answer: (3,-2) -3y = x+2 x+3y = -2 Answer: {(x,y): x + 3y = - 2} Since -3y = -3y is a true statement, the solution to this system is the equation of either one of our given lines.
Ellie wants to give her granddaughter a present of $500 saved in bank account. If Ellie puts $350 in the account which earns an 8% annual interest rate, how long must she wait before the account is worth $500?
The interest amount is $150, which is the difference between the desired amount and what Ellie originally deposits. You can use the interest formula: I = Prt, where I is interest, P is principal, r is rate, and t is time. Just plug in I = 150, P = 350, and r = 0.08, and solve for t: 150 = 350(0.08)t 28t = 150 t = 5.36 years, or approximately: 5 years 4 months.
Times Word Problem
The radar system beeps once every second. How many times will it beep in 3 days? There are 60 seconds in a minute and 60 minutes in an hour. So every hour has: 60 × 60 = 3,600 seconds There are 24 hours in a day, so each day has: 3,600 × 24 = 86,400 seconds Multiply this by 3 to find the number of seconds in 3 days: 3 × 86,400 seconds
Conjugates
The sum and the difference of the same two terms Expression: Its Conjugate: x2 − 3 ⇒ x2 + 3 a + b ⇒ a − b a − b3 ⇒ a + b3 Note: The first poisition in each binomial will be filled by factors of x^2 that are the same. The second position in each binomial will be filled by factors of 4 that are the same.
A plane takes off from New York with 72 of the 90 seats occupied. What percentage of seats are not occupied?
Total - occupied = not occupied. 90 - 72 = 18. Percentage = (Not Occupied Seats)/(Total Seats) x 100 = 18/90 x 100 = 20%
Parallel Lines Vocabulary
Transversal: a line that intersects two or more coplanar lines at different points. Exterior angles: Interior angles: Alternate Interior Angles: Same-Side Interior Angles: Corresponding Angles:
Fractions and Percents
Write the following percent as a fraction in lowest terms. 40% 40/100 2/5 Write the following fraction as a percent. 3/20 3/20 = n/100 (3)(100) = (20)(n) n = 15%
Writing and Solving Multi-Step Equations
Write the following sectence as an equation and solve. 3 times 2 more than x is 27 3(x+2) = 27 3 times 4 less than x is 15 3(x-4) = 15 The sum of 4 times a number and 6 times a number is 50. 4x + 6x = 50 When 19 is subtracted from twice a number, the result is 7. 2x - 19 = 7 6 times a number increased by 8 is 32. 6n + 8 = 32
Composite Number
a number that has more than two factors
Segment Addition Postulate and Midpoint
AB + BC = AC 6x + 5 +4x = 45 x = 4 Congruent Segments: Segments that have the same length Definition of a Midpoint: - If M is the midpoint of segment LN, then segment LM = MN. - If segment LM = MN, then M is the midpoint of segment LN. In the given diagram, Y is the midpoint of segment XZ. Find XZ. XY = YZ 3(x-2) = 5x - 12
Mixture Word Problems
Amount of solution * % of "subtance" = Amount of salt Example: Amount of solution * % salt = Amount of salt Note: Percentage will always be subtracted versus mixture will always be additional x * 0 = 0 Can time both side by 10 or 100 to get a whole number depending what you need to get a whole number. You just got to make the judgement.
Multiplying and Dividing Integers
Any positive or negative divided by zero is undefined (also note that 0 * 0 = 0 and 0 / 0 is undefined). (0) * (-5) = 0 (+8) * (0) = 0 (0) / (-3) = 0 (-6) / (0) = undefined
Integer
Are whole numbers which included negative numbers and or positive number ... but no fractions, decimals, and or other symbols.
Introduction to Proportion
Determine whether the following pair of ratios forms a proportion. 2/7 and 12/42 Do Cross Multiple (2)(42) = (7)(12) 84 = 84 Yes 24 days/8 months and 15 months/5 days (days)(days) ≠ (months)(months) No ^Note: When we do cross multiple, we need to make sure our cross product contain one of each unit. 15ft/9ft and 25sec/15 sec 225 = 225 Yes
Number Word Problems
Don't get intimidate with word problem by looking at the information at once. The best way to solve a word problem is to break it down by segment by segment. One number is 4 times as large as another. Their sum is 45. Find the numbers. x (smaller) 4x (larger) x + 4x = 45 x = 9 Answer: 9 36
Factoring Trinomials with Negative Constants
x^2 - 4x - 3 (x )(x ) [unfactorable]
Slope Formula
m = (y2 - y1) / (x2 - x1) Note: Remember that a negative divided by a negative is a positive (ex) -7/-4 = 7/4 Remember that zero divided by any positive or negative number is zero (ex) 0/2 = 0 Remember that any number divided by zero is undefined, so we say that this line has no slope (ex) -2/0 (undefined) Remember to reduce if possible as your last step (ex) -15/6 = -5/2 When you end up with a positive over a negative in your slope, change it to a negative over a positve (ex) 1/-5 = -1/5 * A common mistake made by students is to forget whether the y's go on top or the x's go on top in the slope formula. If you remember, however, that slope means rise/run, it should make sense that m = (y2 - y1) / (x2 - x1) and not m = (x2 - x1) / (y2 - y1). Why? y represents up/down or "rise," and x represents left/right, or "run."
Isosceles Triangle Theorems
If two sides of a triagnle are congruent, then the angles opposite those sides are congruent. If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
Interest
Interest Formula Interest = Principal * Rate * Time I=Prt I = Amount earned in interest P = Amount invested r = % interest rate per year t = Number of years money is invested How much interest is earned in 5 years on $3,000 invested at an interest rate of 9% per year? I = (3,000)(.09)(5) Interest = $1,350
The Product Rule
x^3 * x^2 = x^5 p * p^6 = p^7 (2a^4)(5a^7) = 10a^11 x^2a + 1 * x^a - 6 = x^3a - 5 (When multiplying two powers together that have the same base, simply add their exponents)
The Quotient Rule
x^5/x^2 = x^3 6a^7/2a^3 = 3a^4 n^2/n^8 = 1/n^6 x^a + 2/ x^a - 1 = x(a+2) - (a-1) = x^a + 2 - a + 1 = x^3
Slope of a Line
Slope = Rise/Run = change in Y/Change in X If you're reading slope from left to right, it depend whether point A is going to be positive or negative to point B. m (slope) = #/# Note: - Most horizontal or flat line is always "zero" (ex) 0/2. - Most vertical or straight up and down line will always have "no slope" or "undefine" (ex) 2/0. - What is the slope of the x-axis? The y-axis? x-axis = (m=0) y-axis = (no slope)
Sales Tax
The sales tax rate in Iowa City, Iowa is 7%. How much tax is charged on a graphing calculator that costs $82? What is the total price? *purchase price * sales tax rate = sale tax 82 * .07 = $5.74 *purchase price + sales tax = total price 82 + 5.74 = $87.74 The sales tax is $3.60 on a concert ticket that costs $45. What is the sales tax rate? *purchase price * sales tax rate = sales tax (45)(r) = 3.60 .08 * 100 r = 8% The sales tax on a dinning room table is $9.75 and the sales tax rate is 6.5%. What is the pruchase price? *purchase price * sales tax rate = sales tax (p)(.065) = 9.75 p = $150
Word Problems
There are two pizza ovens in a restaurant. Oven #1 burns three times as many pizzas as Oven #2. If the restaurant had a total of 12 burnt pizzas on Saturday, how many pizzas did Oven #2 burn? For every x pizzas burnt by Oven #2, Oven #1 burns 3x pizzas. It is given that 12 total pizzas are burnt, and the total number of burnt pizzas is a combination of pizzas burnt by Oven #1 and Oven #2: x + 3x = 12 4x = 12 x = 3
Percent Word Problems
x = 24/100 * 175 Answer: 42 60/100 * x = 15 Answer: 25 x/100 * 25 = 36 Answer: 144
Using Slope to Graph a Line
(3, 1)[plot point A]; m = 3/2[Go by point A, rise/run and plot point B] and then make a line from point A & B. y - intercept = 4 (plot point A on y axis line) m = -2/3 (Go by point A, rise/run and plot point B) (4, -1)[plot point A]; m = -1[Change this integer number into a fraction by just putting number 1 under it (ex) m = -1/1 and then use the rise and run method to from point A and find point B] If a line has a y-intercept of 3 and a slope of -3, what is its x-intercept? Hint: Graph the line. x-intercept = 1 (Note: you won't always be able to determine x-intercept but this one worked out nicely.
Difference of Two Squares
(ex1) a^2 - b^2 Answer: [(a + b)(a - b)] Note: The difference of two squares will always factor as a the product of two conjugates. (ex2) x^2 - 144 Answer: [(x+12)(x-12) (ex3) 36y^2 - 1 Answer: [(6y + 1)(6y - 1)] Note: When factoring, it's critical to be able to recognize when you have the difference of two squares. Students often forget that 1 is a perfect square.
The Power Rule
(x^3)^2 = x^3 * x^3 = x^6 (2a^5)^3 = 8a^15 (3m^5 n)y^2 (2mn^4)^3 = 72m^13 n^14 (ab)^5 = a^5 b^5 (x^2)^a+1
A machine prints 30 newspapers in 20 minutes. At this rate how many hours would it take to print 900 newspapers?
10 If the machine prints 30 newspapers in 20 minutes, it prints: (30 newspapers)/(20 minutes) = 3/2 newspapers per minute. To calculate the time to print 900 newspapers: (900 newspapers)/(3/2 newspapers per minute) = 1800/3 = 600 minutes ÷ 60 minutes per hour = 10 hours
Factor the following trinomial
11x^2 + 3x - 8 (11x + 8)(x - 1) (11x - 1)(x + 1) (11x - 8)(x + 1) = 11x + -8x = 3x (11x + 1)(x - 8) Note: Since the coefficient of the middle term in the original trinomial is odd, you can eliminate pairs of even factors from your list of possibilities.
What is twelve percent as a fraction?
12% = 12/100 = 3/25
Solve for the value of x in the following system: 2x + 4y = 15 3x - 4y = 10
2x + 4y = 15 3x - 4y = 10 ------------- 5x = 25 ------------- x = 25 To solve this system, we will first add the two equations together in order to eliminate the y term. This gives us 5x = 25. Solving for x, we find that x is equal to 5.
Greatest Common Factor (GCF)
30, 42 Answer: 6 x^5 y^3 z^2 , x^3 z^4 Answer: x^3 z^2 The GCF uses the smallest power on each variable. 24a^3 b, 16a^2 b^2 Answer: 8a^2 b 88x^6 y^4 z^2 , 28x^3 y^3 , 30y^2 z Answer = 2y^2
Converting to Slope-Intercept Form and Graphing
4x + 7y = 21 Try to make y = mx + b Why should it make sense that the equation of a vertical line cannot be written in slope-intercept (y = mx + b) form? Answer: The equation of a vertical line cannot be written in slope-intercept (y = mx + b) form, because a vertical line has no slope. Also notice that the vertical line shown on the graph has no y-intercept either.
Adding Radicals and FOILing with Radicals
5√18 + 3√8 - √50 = Make sure we get the radicand to the same number of all the radicand given = 15√2 + 6√2 - 5√2 = add the term together now = 16√2 (7+4√5)(2-3√5) = 14 - 21√5 + 8√5 - 12√25 = 14 - 13√5 - 60 = - 46 - 13√5 7√11(8-√5) = 56√11 - 7√55 Note: Since these radicand are different we cannot combine the number we therefore we have our radicand.
Factoring out the Greatest Common Factor
6x + 4 Answer: 2(3x + 2) 9x^3 + 3x Answer: 3x(3x^2 + 1)
Writing Equations of Lines
A line has a slope of 5/4 and passes through the point (-8, -9). Use a graph to write its equation in slope-intercept form. Answer: y = 1/2x + 3 A line has a slope of 2 and passes through the point (2, 6). Use a graph to write its equation in slope-intercept form. Answer: y = 2x + 2 Note: On this problem you will be going up with the slope, but if you want to subtract the slope so the line can intersect with the y-intercept, all you have to do is subtract rise/run oppositely until the line intersect with the y intercept.
Binomial
A polynomial with two terms 3x^2 + 2
Rational Expression
A ratio of two polynomial expressions It is "Rational" because one is divided by the other, like a ratio. Note: the polynomial we divide by cannot be zero.
Proportion Word Problems
A student must solve 56 problems on a worksheet. She can solve 4 problems every 3 minutes. At this rate, how many minutes will it take her to solve all 56 problems? 4 problems/3 minutes = 56 problems/x minutes (4)(x) = (3)(56) x = 42
Discount
A tennis racket marked at $110.50 is on sale at 20% off. What is the discount? What is the sale price? *original price * rate of discount = discount 110.50 * 22.1000 Discount: $22.10 *original price -discount = sale price 110.50 - 22.10 = 88.40 Sale price: $88.40 An old video game is marked down from $48 to $18. What is the rate of discount (same as finding the percent decrease of the price)? *amount of change/original number 48 - 18/ 48 = 30/48 = .625 Answer: 62.5% discount
Prime Number
A whole number that has exactly two factors, 1 and itself.
Feet Word Problems
Aisha wants to paint the walls of a room. She knows that each can of paint contains one gallon. A half gallon will completely cover a 55 square feet of wall. Each of the four walls of the room is 10 feet high. Two of the walls are 10 feet wide and two of the walls are 15 feet wide. How many 1-gallon buckets of paint does Aisha need to buy in order to fully paint the room? The total number of buckets necessary will be the total area of the walls divided by the total area covered by each bucket. First, calculate the area of the walls Aisha wants to paint. Two of the walls are 10 × 10 and two of the walls are 10 × 15: 2 (10 × 10) = 200 sq. ft. 2 (10 × 15) = 300 sq. ft. So the total square footage of the walls is 500. If a half gallon of paint will cover 55 square feet, then each gallon will cover 2 × 55 = 110 square feet. Four gallons can only cover 440 square feet. Five gallons will cover 550 square feet, which will be enough for the entire area of the walls.
Irrational Number
An Irrational Number is a real number that cannot be written as a simple fraction. Irrational means not Rational Example: π (Pi) is a famous irrational number. Pi π = 3.1415926535897932384626433832795... (and more) We cannot write down a simple fraction that equals Pi. The popular approximation of 22/7 = 3.1428571428571... is close but not accurate.
Polynomial
An expression that can have constants (like 4), variables (like x or y) and exponents (like the 2 in y2), that can be combined using addition, subtraction, multiplication and division, but: • no division by a variable. • a variable's exponents can only be 0,1,2,3,... etc. • it can't have an infinite number of terms. Polynomial comes from poly- (meaning "many") and -nomial (in this case meaning "term") ... so it says "many terms" Constants (like 3, −20, or ½) Variables (like x and y) Exponents (like the 2 in y^2), but only 0, 1, 2, 3, ... etc are allowed *Conclusion: A polynomial can have constants, variables and exponents, but never division by a variable.
Angle Addition Postulate and Angle Bisector
Angle Addition Postulate: m∠1 + m∠2 = 180° Note: start by looking at the interior ray of the angle given. Congruent Angles: angles that have the same measure Adjacent Angles: angles that share a common vertex and side, but no common interior points. (Remember that adjacent angles must share no common interior points.) Angle Bisector: A line that splits an angle into two equal angles. ("Bisect" means to divide into two equal parts.)
Angles and Measure
Angle: figure formed by two rays that share a common endpoint. Endpoint = vertex An angle that measures between 0 degree and 90 degree is called an acute angle. An angle that measures 90 degree is called a right angle. An angle that measures between 90 degree and 180 degree is called an obtuse angle. An angle that measures 180 degree is called a straight angle.
Verticle Angles
Angles that share a common vertex and whose sides form opposite rays. Using the diagram below, find x, m∠POS and m∠QOR. 2x + 5 = 7x - 40
Rational Number
Based on or in accordance with reason or logic. (of a number, quantity, or expression) expressible, or containing quantities that are expressible, as a ratio of whole numbers. When expressed as a decimal, a rational number has a finite or recurring expansion. A Rational Number can be written as a Ratio of two integers (ie a simple fraction).
Triangle Vocabulary and Triangle Sum Theorem
Classifying Triangles no sides = Scalene Triangle 2 sides = Isosceles Triangle 3 sides = Equilateral Triangle 3 acute ∠'s Acute Triangle 1 obtuse ∠ Obtuse Triangle 1 right ∠ Right Triangle All 3 ∠'s Equiangular Triangle
Segments, Rays, and Length
Collinear: When three or more points lie on a straight line. (Two points are always in a line.) A segment is defined as a figure that is composed of two points, in this case A and C, and all the points in between. Points A and C are the endpoints of segments AC. Ray: A part of a line with a start point but no end point (it goes to infinity). Line BA has an endpoint at B, and goes in the direction of A. Line BC has an endpoint at B, and goes in the direction of C. Line BA and Line BC are called opposite rays. Opposite rays are two rays that share a common endpoint, in this case B, have no other points of intersection, and all points on the two rays are collinear. Note that line AC and line CA are no opposite rays, because they do not share a common endpoint. Note opposite rays (all points are not collinear) with angles. The length of a segment can be found by taking the greater endpoint coordinate minus the lesser endpoint coordinate. AG = 2 - (-4) = 6 (segment vs. length) Remember that the coordinate of a given point is the number that is paired with that point.
Combining Like Terms
Combining two or more like terms simplifies an expression by summing constants and summing those variable terms in which the same variables are raised to the same power. 3x * 2 = 6x 3x + 2 3x + 2x = 5x
Complementary and Supplementary Angle
Complementary Angle: If two angles are complementary, then the sum of their measures is 90 degrees. Supplementary Angle: If two angles are supplementary, then the sum of their measures is 180 degrees. If the ratio of an angle to its complement is 2:3, find the measure of each angle. 2x > angle (original) = 36 degree 3x > complement = 54 degree 2x + 3x = 90 5x = 90 x = 18
Consecutive Integer Word Problems
Consecutive Integers 5 > x 6 > x + 1 7 > x + 2 Consecutive Even Integers 10 > x 12 > x + 2 14 > x + 4 Consecutive Odd Integers 7 > x 9 > x +2 11 > x +4 Problem 1: Twice the greater of two consecutive integers is 12 less than 3 times the smaller. Find the integers. x > smaller x + 1 > larger 2(x+1) = 3x-12 Problem 2: The sum of three consecutive odd integers is 111. Find the integers. x > first x + 2 > second x + 4 > third x + x + 2 + x + 4 = 111 Problem 3: The sum of 3 consecutive odd inegers is 36. Find the integers. x > first x + 2 > second x + 4 > third x + (x+2) + (x+4) = 36 x = 10? no solution Since you're asked to find odd integers and x turns out even, there is no solution to this problem.
Introduction to Ratios
Ratio can be written in 3 different ways. (e.g.) 3 to 1 3:1 3/1
F.O.I.L
F.irst O.uter I.nner L.ast (2x + 3)(x - 5) = 2x^2 + -10x + 3x + -15 3xy^2 (Monomial = 1 term) 5x-1 (Binomial = 2 terms) 3x + 5y^2 - 3 (Trinomial = 3 terms) Conjugates (when you multiply two conjugates together, the middle term will always cancel each other out): (p+6)(p-6) p^2 - 6p + 6p - 36 p^2 - 36
Equal Ratios
Find 2 two different ratios that are equal to the following ratio. 4/5 Note: can be times by 2, 3, etc. or divide if can divide to get a whole number. Once you divide, you can use that fraction times by 2. Answer: 8/10, 12/15
Percent Increase or Decrease
Find the percent increase or decrease The price change from $60 to $39. *amount of change/ original number 60-39/60 = 21/60 .35 * 100 Answer: 35% decrease Find the percent increase or decrease. A number changes from 110 to 440. *amount of change/ original number 440-110/ 110 = 330/110 = 3 3 * 100 = 300 300% increase
Translating English to Algebra
First find/define the varible for the following situation, and then write an expression that represents the situattion. 18 more jelly beans are placed in the jar. j = original number of jelly beans in the jar. j+18
Chad's recipe for chocolate chip cookies makes 24 servings of 375 calories each. Chad decided to make 150% of the amount in the recipe rather than the usual 100%. Approximately how many calories are in Chad's batch of cookies?
First, find out how many total calories are in 100% of the recipe: 24(375) = 9,000 calories To find how many calories are in 150% of the recipe, use this equation: x = 9,000 (1.50) x = 13,500
Age Word Problems
Five years ago, Amy was three times as old as Mike. If Mike is 10 years old now, how old is Amy? Set up a table comparing the situation five years ago to the situation today to keep things organized. Name / (Five years ago) / (Today) Mike/ 10 - 5 = 5 / 10 Amy/ 3 x 5 = 15 / 15 + 5 = 20 Mike is 10 years old now, so he must have been 5 years old five years ago. Amy was 3 times as old as Mike five years ago, so Amy must have been 15 five years ago. Add 5 years to Amy's age five years ago to calculate her age now. Amy is 15 + 5 = 20 years old now.
Numerical Bases and Exponents of Zero
Multiplication: 1) 2^3 * 2^2 = 2^5 (Base) 2) 2x^3 * 2x^2 = 4x^5 (Coefficient: a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (e.g., 4 in 4x^ y) Quotient: 3) 2^5/2^2 = 2^3 4) 2x^5/2x^2 = x^3 2^3/2^3 = 2^0 = 1 (Remember that as long as your base is not zero, anything to the zeroeth power is equal to 1)
Solving and Grahping Inequalities
Note: Remember that when you multiply or divide both sides of an inequality by a negative number, you must switch the direction of the inequality sign. When you mutiply or divide both sides by a positive you do not need switch the inequality. Greater than or less than and equal to = solid dot Greater than or less than = open circle
Number and Value Word Problems
Number of Coins * Value of Each = Total Value (Ex1 [Quarters]) 2 * 25 = 50 (Ex2 [Dimes]) 2 * 10 = 20 (Ex3) Student tickets cost $4 and adult tickets cost $7. If the theatre sold 150 tickets and the value of the student tickets was $17 greater than the value of the adult tickets, how many of each type of ticket were sold? Answer: 97 student tickets/53 adult tickets.
Factors
Numbers we can multiply together to get another number. Example: 2 and 3 are factors of 6, because 2 × 3 = 6 A number can have MANY factors! Example: What are the factors of 12? • 3 × 4 = 12, so 3 and 4 are factors of 12 • 2 × 6 = 12, so 2 and 6 are also factors of 12 • and 1 × 12 = 12, so 1 and 12 are factors of 12 as well So 1, 2, 3, 4, 6 and 12 are all factors of 12 And -1, -2, -3, -4, -6 and -12 also, because multiplying negatives makes a positive. In Algebra factors can be expressions like "x+3" etc Example: (x+3) and (x+1) are factors of x2 + 4x + 3
How to find percentage?
Part/Whole x 100% or Take the original number and divide by the total number and times by 100 to get the percetage.
Distance Problems
Sofía is driving to Texas. She travels at 70 kilometers per hour for 2 hours, and 63 kilometers per hour for 5 hours. Over the 7 hour time period what was Sofía's average speed? Average Speed = Total Distance ÷ Total Time If a car travels 360 kilometers in 5 hours, how many kilometers will it travel in 9 hours when driving at the same speed? First calculate the distance the car travels in 1 hour: 360 km ÷ 5 hours = 72 km per hour Then multiply this number by the total number of hours traveled: 9 hours × 72 km per hour = 648 km Notice that the final calculation yields the correct units; this helps to confirm that the calculation is correct.
Multiplying Radicals
√10 * √15 = √150 Answer: 5√6 2√3 * 3√6 = 6√18 Answer: 18√2
Dividing Radicals
√50x^4 z _____________ √2x^1 = √25x^3 z = 5x√xz 2 __ √2 = 2 * √2 __ √2 * √2 = 2√2 _____ √4 = 2√2 _____ 2 = √2 Note: Never leave a radical in the denominator of the problem. Remember to get rid of the square root of the problem, multiply top and bottom of the fraction by that square root. √6 ____ √10 = √15 ____ √25 5/√10 = 5√10 / 10 = √10 / 2 √28 _____ √12 = √7 ____ √3 = √21 ____ 3
Simplifying Radicals
√72 = 2, 36 = 6, 6 Answer: 6√2 ^3√250a^5b^3 Answer: 5ab ^3√2a^2 √-9 = -3, -3 Answer: (None) ^3√-8 = -2, -2, -2 Answer: -2 Note: Notice that we can take the cube root of a negative number, but we cannot take the square root of negative number.
