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...

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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


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