Number Sense and Operations

Associative Property

(a+b)+c=a+(b+c) for addition (ab)c=a(bc) for multiplication

Equation of a Sphere

(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2 the center of the sphere is the point (a,b,c) and the radius is in r units.

standard form of a circle

(x-h)^2 + (y-k)^2 = r^2 the center of the circle is the point (h,k) radius is r units.

Standard form of an ellipse

(x-h)^2/a^2 + (y-k)^2/b^2 = 1

hyperbola equation

(x-h)^2/a^2 - (y-k)^2/b^2 = 1

(xy)^n =

(x^n)(y^n) When exponent (in this case, n) is outside parenthesis, have to do the power of everything inside the parenthesis ex: (2*6)^2 =((2^2)*(6^2)) =(4*36) =144 ex2: (3*5)^3 =((3^3)*(5^3)) =((27)*(125)) =(3,375)

slope formula

(y₂- y₁) / (x₂- x₁) a ratio of the change in height to the change in horizontal distance.

x^0 =

1

Perfect Trinomial Square

1. write the first and last terms as square x² + 2xy + y² = (x+y)² or x² - 2xy + y² = (x-y)² Ex. x² + 12x + 36 x² + 2xy + y² = (x+y)² Answer: (x+6)² or (x+6)(x+6) Ex. 9a² - 6a + 1 x² - 2xy + y² = (x-y)² Answer: (3a-1)² or (3a-1)(3a-1)

x^-n =

1/x^n

prime number

A counting number greater than 1 whose only two factors are the number 1 and itself. ex. 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.

Solving quadratic equations

A quadratic equation is an equation that could be written as Ax² + Bx + C = 0. To solve a quadratic equation: -Put all terms on one side of the equal sign, leaving zero on the other side -Factor -Set each factor equal to zero -Solve each of these equations Example: Solve for x. x² - 6x = 16 x² - 6x - 16 = 0 (x - 8)(x + 2) = 0 x - 8 = 0 --> x = 8 or x + 2 = 0 --> x = -2 Check by inserting your answer in the original equation.

systems of equations

A set of two or more equations with the same variables.

empty set

A situation in which an equation has no true solution.

consistent system of equations

A system of equations that has at least one solution

Inconsistent system of equations

A system of equations with no solution

Systems of Equations - Elimination Method

Add or Subtract the 2 equations so that one of the variables "disappears" Use elimination to solve the system. x - y = −6 + x + y = 8 = 2x + 0 = 2. Add the equations. Solve for x. 2x = 2 x = 1. Substitute x = 1 into one of the original equations and solve for y. x + y = 8 1 + y = 8 y = 8 - 1 y = 7 x - y = −6 1 - 7 = −6 −6 = −6 TRUE x + y = 8 1 + 7 = 8 8 = 8 TRUE Be sure to check your answer in both equations! The answers check. Answer The solution is (1, 7).

Inequalities

Algebraic statements that have ≠, <, >, ≤, or ≥ as their symbols of comparison. when dividing or multiplying by a negative number you must reverse the sign.

rational numbers

All positive and negative integers, fractions and decimal numbers.

real numbers

All rational and irrational numbers

Proportion

An equation stating that two ratios are equal, such as a/b = c/d what is 25% of 160 ? X/160 = 25/100 X= 40

irrational numbers

Any real number which cant be expressed as a fraction of two integers

Line Equations: Standard Form

Ax +By = C Slope: -A/B Y-Intercept: C/B

Distance, Rate, and Time

Distance = Rate * Time Rate = Distance / Time Time = Distance / Rate "A man has to bicycle 30 miles. He will travel at 6 mph. How long will it take?" The distance is 30 miles and the rate is 6 mph. We will plug these values into the formula, like so. d = r x t 30 = 6 x t t = 5 "A bug travels at a rate equal to 3 inches per hour. How far will it travel in 4 hours?" The rate is 3 inches/hour and the time is 4 hours. We will plug these values into the formula, like so. d = r x t 3 x 4 d = 12 "An asteroid travels 300,000 miles in 4 hours. How fast was it traveling?" The distance is 300,000 miles and the time is 4 hours. We will plug these values into the formula, like so. d= r x t 300,000 = r x 4 r = 300,000/ 4 r = 75000

Multiplying two binomials

Foil first outer inner last (Ax + By)(Cx + Dy) = ACx² + ADxy + BCxy + BDxy²

absolute inequalities

Get the absolute value by itself on the left side. Then check to see if it is a "greater" or "less than" statement Ex. 2 |3x+9|<36 2|3x+9|2<362 |3x+9|<18 −18<3x+9<18 −18−9 < 3x+9−9 < 18−9 subtracting 9 from each −27<3x<9 −27/3 < 3x/3 < 9/3 dividing by 3 for each −9<x<3

Subtraction Property

If a - b, then a + (-b)

least common multiple

LCM. The smallest common multiple of a set of two or more numbers. To find lcm, factor each natural number and identify each prime factor the most number of times it appears in any one of the natural numbers in the set. Find the product of the identified prime factors. Ex. LCM of 3 and 4 is 12. Factors of 3: 0,3,6,9,12 Factors of 4: 0,4,8,12,16

perpendicular lines

Lines that intersect to form right angles a/b = -b/a

GCF of a group of algebraic expressions Monomials and polynomials

Monomials: Find the GCF 6x²y. 14xy². 42xy. 2x²y² 6x²y = 2 * 3 * x * x * y 14xy² = 2*7*x*y*y 42xy = 2*3*7*x*y 2x²y² =2*x*x*y*y 2,x,y -> GCF = 2xy Polynomials: Find the GCF 6x^4 - 12x^3 + 4x^2 (2*3*x*x*x*x) - (2*2*3*x*x*x) + (2*2*x*x) 2,x,x -> GCF = 2x^2

Whole Numbers

Natural numbers ( counting numbers) and zero; 0, 1, 2, 3...

Systems of Equations - Substitution Method

Put one equation into the other equations. Works best when one equations starts with a "y=" or "x=" y=2x+4 3x+y=9 We can substitute y in the second equation with the first equation since y = y. 3x+y=9 3x+(2x+4)=9 5x+4=9 5x=5 x=1 This value of x can then be used to find y by substituting 1 with x e.g. in the first equation y=2x+4 y=2⋅1+4 y=6 The solution of the linear system is (1, 6). You can use the substitution method even if both equations of the linear system are in standard form. Just begin by solving one of the equations for one of its variables.

absolute value

The distance a number is from zero on a number line. ALWAYS POSITIVE or zero. IxI.

greatest common factor

The largest factor that two or more numbers have in common. ex. factors of 12 and 16. 12 - 1,2,3,4,6 16 - 1,2,4,8. the gcf is 4!

mean

The same this as the arithmetic average. Use the formula, mean = sum of all numbers in the set/ quantity of numbers in the set

Natural Numbers

The set of numbers 1, 2, 3, 4, ... Also called counting numbers.

Conditional Inequalities

Those with certain values for the variable that will make the condition true and other values for the variable where the condition will be false 2x-5 < 9 = x < 7

Dividing Polynomials

To divide every polynomial by a monomial, divide by EVERY term of the polynomial. Use Exponent Rules to simplify.

Double inequality

Two inequalities can often be combined into a DOUBLE INEQUALITY or COMPOUND INEQUALITY to indicate that the numbers lie between two fixed values. x > -1 and x < 2 or as -1 < x < 2. there would be 2 dots on the graph.

Finding the Midpoint

When the endpoints are (x₁, y₁) and (x₂, y₂), the midpoint is: ((x₁+x₂)/2), ((y₁+y₂)/2)

Factoring a polynomial

Writing a polynomial as a product of polynomial factors. Example: (x-7)(x+4)

additive inverse

a + (-a) = 0; (-a) + a = 0

Division Property

a / b = a / b = a x b^-1 = a x (1/b)

Ratio

a comparison of two quantities; expressed in one of three ways: a to b; a:b; or a/b. The units in both terms must be identical to have a correct ratio. If it is not possible to convert to the same units, write the expression as a rate, such as miles per hour, or miles/hour. EX. At patties pet store, the ratio of dogs to cats to elephants is 3 to 5 to 7. If there are 135 of these animals in total, how many elephants are at patties store? 3x+5x+7x = 135 15x= 135. x=9 27 dogs, 45 cats, 63 elephants. ratio of 4:6 = 4/6 = 2/3

complex fraction

a fraction that contains one or more fractions in the numerator, the denominator, or both. simplify it by rewriting it as a division problem, or multiply both the numerator and denominator by the least common denominator of the fraction in the complex fraction. First look at the bottom fraction, try to find a common denominator if there are more than one at the bottom. once you do rewrite it. the next step is that you keep the numerator, flip the sign, and the fraction into a multiplication problem.

equation

a mathematical statement that two expressions are equal; may be true or false.

complex number

a number of the form a+bi where a and b are real numbers and i is the square root of -1

square root

a number that when multiplied by itself equals a real number. positive real numbers have exactly one real positive nth root, and n could be even or odd. Every real number has exactly one real nth root when n is odd. Negative numbers only have real nth roots if n is odd.

prime factorization

a number written as the product of its prime factors ex. 12/2 = 6 6/2 = 3 2x2x3 = 12

Cartesian Coordinate Plane

a plane divided into four quadrants by the intersection of the x-axis and the y-axis at the origin (x,y) coordinates Quadrant 1 = (+,+) Quadrant 2 = (-,+) Quadrant 3 = (-,-) Quadrant 4 = (+,-)

Monomials

a single constant, variable, or product of constants and variables, such as 2,2x,x, or 2/x. There will never be addition or subtraction symbols in a monomial. Like monomials have like variables, but they may have different coefficients.

identity

a term whole value or determinant is equal to 1

completing the square

a way to convert a quadratic equation in standard form into perfect square form x² + 2ax + a² = c => (x+a)² = c

Multiplicative Identity

a x 1 = a 1 x a = a

Multiplicative Inverse

a x a^-1 = 1 a^-1 x a = 1

Distributive Property

a(b + c) = ab + ac (a + b)c = ac + bc

additive property

a+0=a; 0+a = a

Commutative Property

a+b=b+a for addition ab=ba for multiplication

Polynomials

an algebraic expression which uses addition and subtraction to combine two or more monomials. Two terms make a binomial; three terms make a trinomial. Degree of a monomial: the sum of the exponents of the variables. Degree of a polynomial: the highest degree of any individual term.

rational expression

an algebraic fraction whose numerator and denominator are polynomials Ex. x²−2x−8 / x²−9x + 20 x²−2x−8 / x²−9x + 20 = (x−4)(x+2) / (x−5)(x−4) we cancel out the (x-4) and get a simplified answer of (x+2) / (x−5)

One variable linear equation

an equation written in the form ax+b=0. a is not equal to 0. Also ax+b=c

Graphing two-variable linear inequalities - Coincident

an infinite number of solutions that satisfy both equations. it is represented by a single line, since all points are in common for both linear equations.

Quadratic Formula

ax² + bx + c =0 x = -b ± √(b² - 4ac)/2a

Sum of Two Cubes

a³+b³=(a+b)(a²-ab+b²)

(a/b)^-1 =

b/a

composite number

counting numbers greater than 1 that are not prime. Note: 1 is neither prime nor composite; 2 is the only prime even number. ex. 4,6,8,9 2x2, 2x3, 2x4, 3x3

equivalent equations

equations that have the same solution

Graphing two-variable linear inequalities- Intersection

exactly one solution that satisfies both equations. It is represented by a single point where the two lines intersect on a graph.

horizontal

having a slope of 0. same distance on x axis

parallel

having equal slopes.

vertical

having no slope. all points on y axis.

percent

hundredth or per hundred. to change a percent to a decimal, divide the number by 100. this is accomplished by moving the decimal point two places to the left . the percent of an amount, P, is the percentage rate, R, times the whole amount or base,B. In other words P=RB. To write a percent as a proportion, use the formula R/100 = partial amount/whole amount

cross products

in a proportion, it is the product of the numerator of the first ratio multiplied by the denominator of the second ratio, and the denominator of the first ratio multiplied by the numerator of the second ratio, or a/b = c/d or ad=bc In a true proportion, the cross products will always be equal.

Closure Property

in addition, a + b is a real number. In multiplication, ab is a real number

Compound Interest Formula

interest that is paid multiple times per year for the amount of the principal plus accrued interest. P= Po(1+r/n)^nt P is the total value of the investment Po is the initial value t is the amount of time in years r is the annual interest rate n is the number of times per year the interest is compounded. Ex. If an amount of $5,000 is deposited into a savings account at an annual interest rate of 5%, compounded monthly, the value of the investment after 10 years can be calculated as follows... P = 5000. r = 5/100 = 0.05 (decimal). n = 12. t = 10. If we plug those figures into the formula, we get the following (note that ^ indicates 'to the power of'): A = 5000 (1 + 0.05 / 12) ^ (12(10)) = 8235.05. So, the investment balance after 10 years is $8,235.05.

simple interest

interest that is paid once per year for the principal amount. the formula is I = Prt, I is the amount of interest P is the principal/ amount borrowed r is the annual interest rate t is the amount of time in years Ex. Alex borrows $1,000 for 5 Years, at 10% simple interest: Interest = $1,000 × 10% x 5 Years = $500 • Plus the Principal of $1,000 means Alex needs to pay $1,500 after 5 Years

Solving One-Variable Equations

multiply all terms by the lowest common denominator to eliminate any fractions. look for addition or subtraction to undo so you can isolate the variable on one side of the equal sign. if the side with just variable terms is polynomial, factor that side so one of the factors is just the variable. divide both sides by the coefficient of the variable. when you have a value for the variable, substitute this value into original equation to make sure you have a true equation. Ex. 3x - 4 = 5 + 4. +4 3x = 9 /3 /3 x= 3

Graphing two-variable linear inequalities - Parallel

no solutions satisfy both equations. It is represented by parallel lines on the graph, since the line never intersect.

greatest common divisor

signified by gcd(m,n), where m and n are both natural numbers, it is the same as the greatest common factor.

Absolute values and inequalities

split the inequality into tow parts, one reflecting the positive value of the inequality and one reflecting the negative value. I ax+b I < c => -c < ax + b < c I ax+b I > c. => ax+b < -c or ax + b > c

three methods of systems of equations

substitution, elimination, and transformation of the augmented matrix.

distance from a line to a point not on the line

the length of the segment perpendicular to the line from the point when the line is in the format Ax + By + C = 0, where A, B, and C are coefficients, use a point (x1,y1) not on the line and apply the formula d= I Ax1 + By1 + C I / square root A^2 + B^2 Ex. Find the perpendicular distance from the point (5, 6) to the line −2x + 3y + 4 = 0, using the formula we just found. = ∣(−2)(5)+(3)(6)+4∣ / √4+9 =3.328 Find the distance from the point (−3,7) to the line y= 6/5x + 2 5y=6x+10 6x−5y+10=0 = ∣(6)(−3)+(−5)(7)+10∣ / √36+25 =∣−5.506∣ =5.506 ​

solution set

the set of all solutions of an equation

0^0

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Finding the DIstance Between Two Points

use the Pythagorean theorem, special right triangles, or the distance formula a^2 + b^2 = C^2 or C = square root a^2 + b^2 to find the distance d = √(x₁-x₂)² + (y₁-y₂)²

weighted mean

weighted values, such as w1,w2,w3,..... are assigned to each member of the set x1, x2,x3,.... Use the formula, weighted mean = W1X1 + W2X2 + W3X3 + ... + WnXn / W1 + W2 + W3 + ... + Wn Make sure there is one weighted value for each member of the set.

Graphing two-variable linear inequalities

when ever you have an inequality using the symbol < or >, always use the dashed line for the graph. if the inequality uses < and equal to or > and equal to, use a solid line. All graphs require one side of the line to be shaded. to determine which side to shade, select any point that is not on the line ( the origin is an easy point to use if it is not on the line) and substitute x and y values into the inequality. if the inequality is true, shade the side with that point. if the inequality is false, shade the other side of the line. Graph the inequality y≤4x−2 This line is already in slope-intercept form , with y alone on the left side. Its slope is 4 and its y-intercept is −2. So it's straightforward to graph it. In this case, we make a solid line since we have a "less than or equal to" inequality. Now, substitute x=0,y=0 to decide whether (0,0) satisfies the inequality. 0 ≤ 4(0) − 2 0 ≤ -2 This is false. So, shade the half-plane which does not include the point (0,0).

integers

whole numbers and their opposites (positive and negative)

line equations intercept form

x/x₁ + y/y₁ = 1, where (x₁,0) is the point at which a line intersects the x-axis, and (0,y₁) is the point at which the same line intersects the y-axis.

(x^a)^b =

x^ab or x ^ (a*b)

(x/y)^n =

x^n/y^n

one variable quadratic equations

x² + bx + c = 0 ex. 3x² + 2x − 8 = 0 = (3x-4)(x+2) x=4/3. x= -2

Difference between two squares

x² - y² = (x+y)(x-y) Ex. (x + 3)(x - 2) x² - 2x + 3x -6 x² + x - 6 Ex. (x - 2)(x + 2) x² + 2x - 2x - 4 x² - 4

perfect cubes

x³ + 3x²y + 3xy² + y³ = (x + y)³ x³ - 3x²y + 3xy² - y³ = (x-y)³

sum of two cubes

x³+y³=(x+y)(x²-xy+y²) Ex. 27p³+q³ 27p³ + q³ = (3p)³ + (q)³ = (3p+q)((3p)² − 3pq + q²) = (3p+q)(9p² − 3pq + q²)

difference between two cubes

x³−y³=(x−y)(x²+xy+y²) Ex. Factor 40u³−625v³ Factor out the GCF from the two terms. 40u³−625v³=5(8u³−125v³) Try to write each of the terms in the binomial as a cube of an expression. 8u³−125v³=(2u)³−(5v)³ Use the factorization of difference of cubes to rewrite. 5(8u³−125v³)=5((2u)³−(5v)³) =5[(2u−5v)((2u)²+10uv+(5v)²)] =5(2u−5v)(4u²+10uv+25v²)

Line Equations: Point-Slope Form

y - y1 = m(x - x1), where m is the slope and (x1, y1) is a point on the line

Line Equations: Point-Slope Form

y2 - y1 = m(x2 - x1), where m is the slope and (x1, y1) is a point on the line

parabola equation

y=ax^2+bx+c a does not equal 0. In this form, if the value of a is positive, the parabola opens upward. If the value of a is negative, it opens downward. the axis of symmetry is the line = -b/2a. the vertex of the parabola is the point (-b/2a, 4ac-b^2/ 4a) the vertex form of a parabola is a(x-h)^2 + k. (h,k)

Line Equations: Slope-Intercept Form

y=mx + b where m is the slope and b is the y-intercept