Equations of Circles, Circle Equations, parabola equations, Coordinate Geometry, Solving Linear Inequalities, Solving Linear Equations Review, GRE - Exponents Rules, GRE Algebra, Absolute Value, GRE - Exponents and Roots, Fraction Rules, Kaplan GRE C...
Factor: 9x^2 - 16 =
(3x - 4) (3x + 4) *Difference of two squares
Square of a difference
(a-b)^2 = a^2 - 2ab +b^2
(x^a)(y^a)
(xy)^a
Which is greater -17 or -13?
- 13
-32 divided by 3
-11 remainder 1 greatest multiple of 3 that is less than or equal to -32 is (-11)(3) or -33, which is 1 less than -32
15 dollars less
-15
1,500 meters below sea level
-1500
20 degrees below zero
-20
|0|
0
factor and multiple of 0
0 is not a factor of any integer except 0; 0 is a multiple of every integer
6 divided by 24
0 remainder 6 greatest multiple of 24 that is less than or equal to 6 is (0)(24) or 0 which is 6 less than 6
factor and multiple of 1
1 is a factor of every integer; 1 is not a multiple of any integer except 1 and -1
factors of 60
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30 and 60 and all their negatives
25 is a multiple of only six integers
1, 5, 25, -1, -5, -25
How to Subtract Mixed Numbers
1. Change fractional parts to equivalent fractions with the LCM as a denominator, if needed. 2. If necessary, borrow from the whole number to subtract fractions. 3. Subtract whole numbers and fractions separately.
How to Add Mixed Numbers
1. Change fractions to equivalent fractions with a common denominator. 2. Add fractions. 3. Add whole numbers. 4. If fraction part is an improper fraction, add to the whole number needed.
Dividing Mixed Numbers
1. Change fractions to improper fractions 2. Find the reciprical of the 2nd number. 3. Change the division sign to a multiplication sign 4. Multiply
How to add fractions with different denominators
1. Find Common Denominators 2. find new numerators (whatever you do to denominator you must do to numerator). 3. Add Numerators 4. Denominator remains the same 5. Simplify
How to Divide Fractions
1. Find the reciprocal of the 2nd number. 2. Change the division sign to a multiplication sign 3. Multiply. 4. Simplify
How to Multiply Fractions
1. Multiply across numerators 2. Multiply across denominators
How to add and subtract fractions with the same denominator
1. Simply add or subtract numerators 2. Keep denominators the same 3. Simplify
How to Multiply Mixed Numbers
1. Write each mixed number as an improper fraction. 2. Then multiply the two improper fractions.
solve for x and y using the elimination method: 2x + 3y = 15 and x + 2y = 11
1. eliminate x by multiplying the second equation by -2 2. the second equation becomes: -2x - 4y = -22 3. add the new second equation to the first one: (-2x - 4y = -22) + (2x + 3y = 15) = - y = - 7 or y = 7 4. solve for x by plugging the value for y into one of the equations: 2x + 21 = 15 2x = -6 x = -3
|5+6|
11
prime factorization examples
12 = (2)(2)(3) = (2 squared)(3) 14 = (2)(7) 81 = (3)(3)(3)(3) = 3 to the 4th power 338 = (2)(13)(13) = (2)(13 squared) 800 = (2)(2)(2)(2)(2)(5)(5) = (2 to the 5th power)(5 squared) 1155 = (3)(5)(7)(11)
1/8
12.5%, 0.125
The absolute value of -3 plus the absolute value of 10
13
1/6
16.6%, 0.166
100 divided by 45
2 remainder 10 the greatest multiple of 45 that is less than or equal to 100 is (2)(45) or 90 which is 10 less than a 100
list of prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89
1/5
20%, 0.2
23 feet above me
23
1/4
25%, 0.25
|2x + 3| = 5
2x + 3 = 5 -- x = 1 Or 2x + 3 = -5 -- x = -4
|-3|
3
1/3
33.3%, 0.333
3/8
37.5%, 0.375
|4|
4
list of composite numbers
4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39...
2/5
40%, 0.4
The absolute value of - 47
47
1/2
50%, 0.5
The absolute value of 52
52
57 inches tall
57
24 divided by 4
6 remainder 0
3/5
60%, 0.6
5/8
62.5%, 0.625
2/3
66.6%, 0.666
3/4
75%, 0.75
4/5
80%, 0.8
5/6
83.3%, 0.833
7/8
87.5%, 0.875
|A| = B
A = B OR A = -B
equation
A mathematical sentence that says two expressions are equal.
Positive Integers
All numbers greater than zero.
Negative Integers
All numbers less than zero.
Integers
All whole numbers including negative and positive numbers.
Squaring multiples of 5
Always ends in -25 (ones and tens places) then multiply the first digit of the multiple of ten above and below (ex. 35^2 => 3*4 = 12 => 1225)
positive number
Any number that is greater than zero.
negative number
Any number that is less than zero.
Positive Integer
Any whole number greater than zero
Negative Integer
Any whole number less than zero
how do you avoid extraneous solutions with absolute values?
Check your work by plugging values into the original equation
0
Find the absolute value.
15
Find the absolute value.
212
Find the absolute value.
35
Find the absolute value.
39
Find the absolute value.
7
Find the absolute value.
Factor: x^7 - 4x^5 =
First factor out GCF: x^5 (x^2 - 4) Then Factor as a difference of two squares: (x^5) [(x - 2)(x + 2)]
Adjacent Squares
For any square (n^2) the next square (n+1)^2 is just equal to n^2 + n + (n+1) (ex. 41^2 => 40^2 = 1600 => 1600 +40 +41 = 1681)
GCF of 30 and 75
GCF is 15 divisors of 30 are 1, 2, 3, 5, 6, 10, 15, 30 divisors of 75 are 1, 3, 5, 15, 25, 75
Raise a power to a power
Keep the base, multiply the exponents
LCM of 30 and 75
LCM is 150 multiples of 30 are 30, 60, 90, 120, 150 multiples of 75 are 75, 150
What reverses the direction of an inequality?
Multiplying or dividing by a negative number
Can you distribute exponents across add/subtract?
NO!!!!
(98)(102) = ? (79)(81) = ? ~ using FOIL method ~
Notice that (98)(102) can be written as (100 - 2)(100 + 2). Now you should see the (x - y)(x + y) form, which expressed correctly is 100^2 - 2^2 = 10,000 - 4 = 9,996. (80-1)(80+1) = 6400 - 80 + 80 - 1 = 6400 - 1 = 6399
square of multiple of 10
Square non zero digits and then add the zeros (ex. 70^2 => 7^2 = 49 => 4900)
fractional exponent
The numerator is the power of the base, the denominator is the type of root
integers
The set of whole numbers and their opposites {. . .-2, -1, 0, 1, 2. . .}.
reciprocal (multiplicative inverses)
Two numbers whose product is one.
The distance a number is from zero on a number line
What is Absolute Value
one solution
When two or more linear equations intersect (cross) at one point.
b)
Which letter graph does this linear inequality represent? Write letter as a) or b) etc.
c)
Which letter graph does this linear inequality represent? Write letter as a) or b) etc.
-2,-7
Which of these numbers are possible values for x? Write commas inbetween the numbers.
25, 2, -8, -13
Which of these numbers are possible values for x? Write commas inbetween the numbers.
x + 9 < 22
Write an inequality: The sum of a number and 9 is less than 22
7 - 2x > 85
Write an inequality: Two times a number is subtracted from 7. The difference is greater than 85
(4)
Write answer as (1) or (2) etc.
1)
Write graph number as 1), 2) etc.
prime factorization
a number written as the product of its prime factors
prime number
a whole number greater than 1 that has two factors, 1 and itself
(a+b)^2=
a^2 + 2ab + b^2
Multiply Exponents with the same base
add the exponents
composite number
an integer greater than 1 that is not a prime number
even integer
an integer that is divisible by 2 {...-6, -4, -2, 0, 2, 4, 6...}
odd integer
an integer that is not divisible by 2 {...-5, -3, -1, 1, 3, 5...}
the discriminant formula
b^2-4ac
ways to prove something is a parallelogram
both pair of opposite sides equal (with distance formula) both pair of opposite sides parallel (with slope formula) diagonals bisect each other (midpoints of diagonals are the same)
(x + 1)² + (y - 5)² = 100
center (-1, 5) radius 10
(x+3)² + (y-6)² = 16
center (-3, 6) radius 4
product of an even integer and an odd integer
even integer
product of two even integers
even integer
sum of two even integers
even integer
sum of two odd integers
even integer
list of nonzero integer multiples
infinite
product of a positive and a negative integer
negative integer
product of two odd integers
odd integer
the sum of an even integer and an odd integer
odd integer
product of two negative integers
positive integer
product of two positive integers
positive integer
the points at which a quadratic equation intersects the x-axis are referred to as:
quadratic roots, solutions, zeros, x-intercepts
sqrt(a) * sqrt(b)
sqrt(a*b)
sqrt(a) / sqrt(b)
sqrt(a/b)
To divide exponents with the same base
subtract the exponents
greatest common factor/divisor
the largest factor that two or more numbers have in common.
least common multiple
the smallest multiple (other than zero) that two or more numbers have in common
vertical lines
undefined slopes (are of the form x=#)
quotient and remainder
when integer c is divided by a positive integer d, you first find the greatest multiple of d that is less than or equal to c that multiple of d can be expressed as the product qd, where q is the quotient the remainder is equal to c minus that multiple of d, or r = c - qd the remainder is always greater than or equal to 0 and less than d
factor/divisor
when integers are multiplied, each of the integer is called a factor or divisor of the resulting product ex. (2)(3)(10) = 60 so 2, 3, 10 are factors of 60
An integer must be a ________________
whole number
integers
whole numbers, their opposites, and zero
|x| = 5
x = -5 or x=5
formula for axis of symmetry
x= -b/2a
(x + y)(x + y) =
x^2 + 2xy + y^2 (factored) (The square of a sum)
(x - 5) (x + 5) = (a + 3)(a + 3) = (b - 2c)(b - 2c) =
x^2 - 25 a^2 + 6a + 9 b^2 - 4bc + 4c^2
(x-y)(x -y) =
x^2 - 2xy + y^2 (factored) (the square of a difference)
(x - y) (x + y) =
x^2 - y^2 (factored)
x^a * x^b
x^a+b
x^a / x^b
x^a-b
(x/y)^a
x^a/y^a
(x^a)^b
x^ab
center (0, 7) radius 13
x² + (y -7)² = 169
vertex form of a quadratic equation
y= a(x-h)^2 + k vertex (h, k) x=h
standard form ( stand firm of a quadratic equation
y= ax2+bx+c
absolute value is the distance from ________ on a number line
zero
horizontal lines
zero slope (are of the form y=#)
10
| 10 | =
26
| 26 | =
PLACE THESE VALUES IN LEAST TO GREATEST ORDER... |25|, |-20|, |-30|
|-20|, |25|, |-30|
PLACE THESE VALUES IN ASCENDING VALUE... |-13|, |5|, |-2|
|-2|, |5|, |-13|
PLACE THESE VALUES IN DESCENDING ORDER |-6|, |12|, |15|, |3|
|15|, |12|, |-6|, |3|
|3x + 2| + 1 = 5
|3x +2| = 4 3x + 2 = 4 -- x = 2/3 OR 3x + 2 = -4 -- x = -2
5
|−5 |=
constant
An unchanging value; a number without a variable.
substitution
In Algebra "Substitution" means putting numbers where the letters are.
No
Is 4 a possible answer for x: 2x > 8
c>-13
Solve for c:
g > 2
Solve for g: 2g + 4.5 > 8.5
x² + y² = 49
center (0, 0) radius 7
parallel lines
have the same slope
Factor: 25x^2 - 64y^2=
(5x + 8y)(5x - 8y) *Difference of two squares
Difference of two squares a^2 - b^2 =
(a + b)(a - b) * The test is in love with this pattern
(a+b)^2
(a+b)(a+b)= a^2 + 2ab + b^2 -- This is called the sum of a square [NOT (a^2 + b^2)]
Square of a sum
(a+b)^2 = (a+b)(a+b) = a^2 + 2ab + b^2
center (-6, 0) radius 2
(x + 6)² + y² = 4
center (1, 0) radius 15
(x - 1)² + y² = 225
center (2, 3) radius 4
(x - 2)² + (y -3)² = 16
center (3, 4) radius 1
(x - 3)² + (y - 4)² = 1
center (-3, -4) radius 1
(x+3)² + (y+4)² = 1
Factor: x^2 - 49
(x-7)(x+7) *Difference of two squares
Factor: x^4 - 81 =
(x^2 + 9)(x^2 -9) = (x^2 + 9)(x + 3)(x - 3)
Factor: x^6 - 16
(x^3 - 4)(x^3 + 4)
Factor: x^8 - 9y^2 =
(x^4 - 3y)(x^4 + 3y)
Factor: x^2y^2 - 1
(xy - 1)(xy + 1) * The difference of two squares
Two methods for dealing with two equations and two unknowns (x, y)
1. Substitution: If one of the variables in one of the equations has a coefficient of +/-1, it is easy to use substitution. Solve for one variable and then plug in to the other equation. 2. If the coefficients are not +/-1, use the elimination method. The strategy is to multiply both sides of one equation by one number and both sides of the other equation by another number so that for one of the variables, the two coefficients are equal and opposite and can then be cancelled.
Procedure for majority of quadratic equations:
1. get everything on one side, set equal to zero 2. divide by any GCF 3. Factor 4. Use the Zero Product Property to create two linear equations and solve
Simplify: 13x^2 + 26x = 15xy - 18xy^2 = X^5 - 8x^4 + 15x^3 =
13x (x + 2) 3xy (5 - 6y) x^2 (x^3 - 8x^2 + 15x)
prime factorization of 1599 =
1599 = 1600 - 1 = (40^2) - (1^2) = (40 + 1)(40 - 1) = (40)(39) = (41)(13)(3) [these are the prime factors] Note: If there is a large 4 digit number on the test, and to solve the problem, you realize that you would need to have the prime factorization of the number, it is a very, it is very likely that the test is expecting you to recognize that that large number can be expressed with the difference of two squares.
Prime Factorization of 2491
2491 = 2500 - 9 = (50 - 3)(50 + 3) = (47)(53) [This is the prime factorization] Note: If there is a large 4 digit number on the test, and to solve the problem, you realize that you would need to have the prime factorization of the number, it is a very, it is very likely that the test is expecting you to recognize that that large number can be expressed with the difference of two squares.
Prime Factorization of 9975
9975 = 10000 - 25 = (100 - 5)(100 + 5) = (95)(105) = (5*19)(5*21) = (5 * 19) (5 * 7 * 3) = (5^2 * 3 * 7 *19) [These are the prime factors] Note: If there is a large 4 digit number on the test, and to solve the problem, you realize that you would need to have the prime factorization of the number, it is a very, it is very likely that the test is expecting you to recognize that that large number can be expressed with the difference of two squares.
algebraic expression
A mathematical expression that consists of variables, numbers and operations
variable
A symbol used to represent an unknown quantity (usually a letter).
solution
A value or values which, when substituted for a variable in an equation, make the equation true.
Distributive Property
A(B + C) = AB + AC
infinitely many solutions
If the variables disappear, and you get a statement that is always true, such as 0 = 0 or 3 = 3, then there are "infinite solutions", meaning, when graphed, the two equations would form the same line
no solution
If the variables disappear, and you get a statement that is never true, such as 0 = 5 or 4 = 7 then there is "no solution", meaning, when graphed, the two equations would form parallel lines, which never intersect.
inverse operations
Inverse operations are opposite operations that undo each other. Addition and subtraction are inverse operations. Multiplication and division are inverse operations.
b≤5
Solve for b:
t≤9/8
Solve for t (leave as a fraction):
t≤-5
Solve for t:
x < 2
Solve for x: 6(8x - 14) < 12
Negative Exponent Property
Take the multiplication inverse or reciprocal of the power. Repeated division.
like terms
Terms whose variables are the same. Example: 7x and 2x are like terms because the variables are both "x" BUT, 7x and 7x^2 are NOT like terms (they are Unlike Terms)
coefficient
The number multiplied by a variable to indicate "how many" of the unknown value exist.
(x + 6)² + y² = 13
center (-6, 0) radius √13
(x + 7)² + (y + 5)² = 9
center (-7, -5) radius 3
x² + (y - 2)² = 25
center (0, 2) radius 5
(x-2)² + y² = 25
center (2, 0) radius 5
ways to prove something is a rectangle
has a right angle (opposite reciprocal slopes) or diagonals are congruent
perpendicular lines
have opposite reciprocal slopes
altitude of a triangle
joins a vertex PERPENDICULAR to the opposite side
median of a triangle
joins a vertex to the MIDPOINT of the opposite side