SAT Math Formulas

Probability Equation

# of desired outcomes/ # of total outcomes

factor

# that divdes in to another # evenly without a remainder Ex: the factors of 52 are 1, 2, 4, 13, 26, and 52

If u+t= 5 and u-t =2, what is the value of (u−t )( u²-t²) ?

( u²-t²)=(u+t)(u-t) therefore (u−t )( u²-t²)= (u-t)(u+t)(u-t)= (2)(5)(2)=20

Length of an arc

(2πr)(central angle/360)

(a+b/2)²

(a+b/2)(a+b/2)

Complex Conjugate (a+bi)(a-bi)

(a+bi)(a-bi)= a²+b²

sum of terms

(average)(# of terms)

Area of a Triangle

(b*h)/2 or 1/2 *b* h

Convert degrees to radians

(degrees)(π/180)

Percent Increase Equation

(final - initial/initial)*100

Sum of Interior Angles Equation

(n-2)180 n= # of sides of a polygon

Convert radians to degrees

(radians)(180/π)

ax³ bx² cx + d = 0 If the equation has roots −1, −3, and 5, which of the following is a factor of ax³ bx² cx + d = 0 ?

(x+1), x+3) (x-5)

Equation of a Circle

(x-h)² + (y-k)² = r² (h, k)Conveting = point for center of circle r= radius

Exponent Rule (xy)²

(xy)² = x²y²

Exponent Rule (power raised to a power) (x³)²

(x³)²= 3 * 2 = x⁶

Average Growth rate

(y2-y1)/(x2-x1) Change in y axis/ change in x-axis

Slope

(y₂-y₁)/(x₂-x₁) rise/run

The graph of y x = 2x² + 10x + 12 is shown. If the graph crosses the y-axis at the point (0, k) , what is the value of k ?

*the last term c = y intercept but also 1)if we plug in 0 for x and solve we get 12 so 12 is the answer *we plug in 0 for x bc y intercept is when x=0

Exponet Rule (-1)ⁿ

+1 if n is even -1 if n is odd

Integers

-3, 0, 2 no fractions or decimals!

Determine the mean from a frequency table Score|Frequency 3 3 4 3 5 4

1) Add all the frequencies to determine N or the total number of frequencies 3+3+4= 10 2) (#)(frequency) + (#)(frequency) + (#)(frequency)/total # of frequencies (3)(3) + (4)(3) + (5)(4) /10 3) solve- 4.1

polynomial long divsion

1) Divide the term with the highest power inside the division symbol by the term with the highest power outside the division symbol. 2)Multiply (or distribute) the answer obtained in the previous step by the polynomial in front of the division symbol. 3) Subtract and bring down the next term. 4) continue until there are no terms t bring doown

The expression 5x−2/x+ 3 is equivalent to which of the following?

1) Do long division

Some values of the linear function f are shown in the table above. What is the value of f( 3)? x f(x) 0 -2 2 4 6 7

1) Look at how the x's and the y's are changing

The line with the equation 4/5x + 1/3y = 1 is graphed in the xy‑plane. What is the x-coordinate of the x‑intercept of the line?

1) Make y=0 bc to solve for the x-intercpet y has to be 0

The radius of a given circle is increased by 20%. What is the percent increase of the area of the circle.

1) Plug in a random value for the radius to see what area is 5→πr²→25π 2) FInd the vlaue that is 20% greater than 5 PLug this in and see what the are is 6→πr²→36π 3) Do final-initial/ initial equation using the difference between the two equation (36π-25π/25π)*100= (11/25*100)= 44

What is the sum of all values of m that satisfy 2m² − 16m + 8 = 0?

1) Simplify m²-8m+4 2) We know its factor will multiply to 4 and add to b b is -8 3) the sum of all values of m will be b but the oppoatie sign so 8

How to complete the Square 5x-4x-2=0

1) Simplify *Leading coefficient must be 1 to complete the square (divide all terms by 5 x²-4/5x-2/5=0 2)Move the only numbers term to the other side of the equation (x²-4/5x=2/5) 3) Do (b/2)² and add it to both sides of the equation [(0.80)/2]²=0.16 (so add 0.16 on both sides 4) Simplify the left side in to a perfect square (x-0.4)² = 0.56 5) Take the square roots on both sides and solve x-0.4= ±√0.56

x/x-3= 2x/2 Which of the following represents all the possible values of x that satisfy the equation above

1) Simplify to get x/x-3= x 2) Clear the fraction x= x²-3x 3) Get the equation to equal 0 x²-4x=0 4)Distribute out common term x(x-4)=0 5) Make both equal to 0 x=0 x-4=0 x=0 x=4

In 2008, there were 21 states with 10 or more electoral votes, as shown in the table above. Based on the table, what was the median number of electoral votes for the 21 states?

1) Total frequency is 21 states 2) The median is the middle # 3) So do 21/2=10.5=11 and then count to the 11th position on the frequency table 4) Get your answer *make sure you're looking at the x column(votes)

How to solve a quadratic equal to 0 Ex: x²+ 4x+3=0

1) factor x²+ 4x+3 (x+3) (x+1)=0 2) set both parts = 0 and solve for x (x+3)= 0 (x+1)= 0 x=-3 x=-1

Simplifying Radicals

1) get prime factors2*7*7 2) write them under the radical√2*7*7 3)simplify the radical (square of a # will always cancel out and come out of the radical as the number)7√2

A book was on sale for 40% off its original price. If the sale price of the book was $18.00, what was the original price of the book? (Assume there is no sales tax.)

1) since the book was on sale for 40% the sale price $18 is 60% of the original price so use the equation 0.60x=18 and solve for x to get the original price

In the xy-plane, the graph of y = 3x² - 14x intersects the graph of y = x at the points (0, 0) and (a, a) . What is the value of a?

1) since we have 2 equations where one equation equal a variable we can use substitution so... 2) plug x in for y in 3x² − 14x 3) subtract the x so the equation can equal 0 to get 0 = 3x² − 14x 4) factor and solve 3x(x-5)=0 x-5=0 x=5 3x=0 x=0

If a⁻½=x, where a > 0, what is a in terms of x ?

1) solve for a in the equation given a⁻½=x 2) cancel out the exponent by raising both sides to the -2 a=x⁻² x⁻²= 1/x²

To solve 2 linear equations Substitution x + y = 3 and 4x − y = 2

1) solve for a variable in one equation y=-x+3 2) substitute that variable in to the other equation and solve 4x - (-x+3)=2 x=1 3) substitute the solved variable back in to one of the original equations to solve for other variable 1+y=3 y=2

The graph of a line in the xy-plane passes through the point (1, 4) and crosses the x-axis at the point (2, 0). The line crosses the y-axis at the point (0, b) . What is the value of b ?

1) use points given to find slope 2) Then solve for y-intercept using slope solved for and one of the original points

Determine the median from a frequency table (odd) Score|Frequency 2 2 3 1 5 4

1)Add all the frequencies to determine N or the total number of frequencies 2+1+4= 7=N 2) SInce odd do N/2 to get the middle position (round to nearest whole #) 7/2= 3.5=4 3) Count to the space you solved for 5 is your answer

Determine the median from a frequency table (even) Score|Frequency 2 2 3 1 4 5

1)Add all the frequencies to determine N or the total number of frequencies 2+1+5= 8=N 2)Since even do N/2 and N/2+ 1 to get the numbers surrounding the middle position 8/2= 4 8/2 +1 =5 3) Count to the spaces you solved for then do the average of them 4 for both of them (4+4)/2=4

Probability Example What is the probability of flipping a coin twice and getting heads two times in a row?

1/2 * 1/2 = 1/4

Probability Equations Example What is your probability that you will chose the right answer on the SAT?

1/5 =20%

Negative Exponent Rules 1/x² x⁻³

1/x² = x⁻² x⁻³ = 1/x³

In 2015 the populations of City X and City Y were equal. From 2010 to 2015, the population of City X increased by 20% and the population of City Y decreased by 10%. If the population of City X was 120,000 in 2010, what was the population of City Y

120,000(1+0.20) =144000 solve for y y(1- 0.10)=144000 y= 160,000

Jenny has a pitcher that contains 1 gallon of water. How many times could Jenny completely fill the glass with 1 gallon of water? ( 1 gallon =128 fluid ounces) *also given the glass is 16foz

16x=128 128/16=8

Circumferecne of Circle

2πr or πd

Special Right Triangle 1

3-4-5 triangle and multiples of this triangle (ex: 6-8-10, 9-12-15)

p(x)= 3x⁴+6x³-5x-10 Determine whether the given binomial (x-2) is a factor of the polynomial p(x)

3x⁴+6x³-5x-10=0 1)Plugin the opposite of the number in the binomial into the equation 2 2) if the polynomial equals 0 then the binomial is a factor

Difference of Squares Example 4x² - 81y²

4x² - 81y² (2x)² - (9y)² (2x+ 9y) (2x - 9y)

Fundamental Counting Principle Example Mark has 5 pants and 7 shirts in his closet. He wants to wear a different pant/shirt combination each day without buying new clothes for as long as he can. How many weeks can he do this for?

5 *7 = 35 days or 5 weeks

Special Right Triangle 2

5-12-13 and multiples of this triangle

Set up the Equation to solve A group of friends decided to divide the $800 cost of a trip equally among themselves. When two of the friends decided not to go on the trip, those remaining still divided the $800 cost equally, but each friend's share of the cost increased by $20. How many friends were in the group originally

800/x +20 = 800/x-2

Compound Interest

A= P(1+r/n)ⁿ⁺ P= initial or starting amount r= interest rate(decimal) t= time n= the # of times the interest compounded

Simple Interest

A= Prt p=principal amount = initial or starting amount amount (borrowed or invested) r = interest rate (expressed as decimal) t = time

Prime Factorization

Breaking up of a # in to its prime factors use factor tree to do this

How do you figure out many zeroes a function has?

By looking at how many times the function touches the x-axis

The peregrine falcon can reach speeds of up to 200 miles per hour while diving to catch prey, making it the fastest animal on the planet when in a dive. What is a peregrine falcon's maximum speed while diving to catch prey, in feet per second? (Round your answer to the nearest whole number. 1 mile = 5280 feet)

Cancel out all units except units in final answer Convert m/hr to ft/sec (200mi/hr) (5280ft/1 mi)(1hr/60m) (60m/3600sec)

Geometric Sequence

Each term is equal to the previous term times r r = 2 and t1 = 3 gives the sequence 3, 6, 12, 24

Arithmetic Sequence

Each terms is equal to the previous term plus d Ex: d = 4 and t1 = 3 gives the sequence 3, 7, 11, 15

The graph of a line in the xy-plane has slope 2 and contains the point (1, 8). The graph of a second line passes through the points (1, 2) and (2, 1). If the two lines intersect at the point (a, b) , what is the value of a+b ?

For Both Equations 1) Find slope 2)Find y intercept using a point 3) Make equations equal to each other and solve for x 4) plug x back in to one of the original equations and solve for y 5) Figure out the value of x+y or a+b

Fundamental Counting Principle (can also use for combination problems)

If an event can happen in N ways and another independent event can happen in M ways then both events can happen in N* M ways

Complementary angle property of sine and cosine

In a right triangle the sine of one angle=the cosine of the other acute angle since these acute angles add to 90 we know the acute angles are complementary

Area of a Rectangle

Length* width

Volume of a rectangular solid

Length*width*height

Orders of Operations

PEMDAS

Elimination 3x + 4y = 52 5x + y = 30

The canceling out of a variable by adding or subtracting equations 1) line up same variables and figure out which variable to cancel 3x+4y=52 −4(5x+y)=−4(30) 2) Cancel out a variable by adding them together x+4y=52 −20x-4y=−120 3) Solve for left over variable −17x=-68 x=4 4) Substitute answer in to one of the original equations and solve for the other variable

H = 1.88L + 32.01 The formula above can be used to approximate the height H, in inches, of an adult male based on the length L, in inches, of his femur. What is the meaning of 1.88 in this context?

The number being multiplied by the variable is always ether showing an increase or decrease in what the equation is equaled to think of y =mx+b m=y/x!! The approximate increase in a man's height, in inches, for each one-inch increase in his femur length

SAT problem If aⁿ÷⁴ = 16 for positive integers a and n, what is one possible value of n?

When bases are the same you can make their exponents equal 1)Get same base on both sides 2ⁿ÷⁴ = 2⁴ 2) make exponents equal to each other n÷4= 4 3) solve n

When adding complex numbers Ex: (2+3i)+ (4+8i)

add things in the same family ex: 2+4 = 6 3i+8i= 11i so 6+11i

Difference of Squares a² + 2ab + b² a² - 2ab + b²

a² + 2ab + b² =(a+b) (a+b) a² - 2ab + b²= (a-b)(a-b)

Difference of Squares a² - b²

a² - b² = (a+b) (a-b)

Vertex form y=-a(x-b)²+c what is the vertex

b, c

Rule of discriminant when determining type of solution

b²-4ac= positive then ther will be 2 real solution b²-4ac=0 then 1 real solution b²-4ac= negative then no real solution

Quadratic Rules of c and vertex (parabola) y=ax²+bc+c

c= y-intercept the x coordinate of vertex= -b/2a

Density equation

d=m/v

When asked too look at standard deviation of number in a table Look at...

each numbers distance between the mean

is/are means

equal sign

Each successive year 1% of the current value is added to the value of the account

exponential growth

hamburger has 50 more calories than each order of fries means

h=f+50

inscribed angle in a circle is equal to

half its arc measure

Which of the following complex numbers is equivalent to 3-5i/8+2i?

i CAN NOT be in the denominator Multiply fraction by the CONJUGATE of the denominater (8-2i) 3-5i/8+2i *8-2i/8-2i

What is i i² i³, and i⁴?

i= √-1 i²= -1 i³ = -i i⁴= 1

Quadratic Rules of a (parabola ) y=ax²+bc+c

if a>0 parabola opens up if a<0 parabola opens down

How do you know if a system of linear equation has infinite solutions

if the euqations for the lines are the same

Real #s

integers, fractions, deciamls, and irrationals(π √2)

if function is raised to an even power (like x²) then

it will NOT cross the x axis

if function is raised to an odd power (like x³) then

it will cross the axis

central angle in a circle is equal to

its arc measure

Area of a Parallelogram

length*height

SAT Problem How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to obtain a 15% saline solution?

liters of saline solution × percent of saline solution x=liters 3(0.10) + 0.25x = 0.15(x + 3) the final amount of liters has to be added to the 3 liters already there

What is the set of all solutions to the equation? √2x+1= -x

make sure that all answers satisfy the equation by plugging answers back in to the equation 1) solve to get -1 and 2 2) Plug back in √2(2)+2 = -2 doesnt work so not an answer √2(-1)+2 =-(-1) does work so final answer

median

middle value when the #s are ordered

multiple

multiples of a # are divisible by that # without a remainder the positive Ex: multiples of 20 are 20, 40, 60, 80,

of means per means

multiplication division

anytime you divde or multiply a # by a negative # in a inequality you...

must flip the sign

Equation for Diagonal in regular polygon Example hexagon

n= # of sides of polygon d= [n(n-3)]/2 d= [6(6-3)]/2 d= 9 sides

Intersecting Lines

opposite angels are equal

Percent Equation

part= (Percent/100) * whole

Percent Equation Example 30 is 20% of what?

part= (Percent/100) * whole 30= (20/100) * x

Percent Equation Example 45 is what percent of 60?

part= (Percent/100) * whole 45= (x/100) * 60

Percent Equation Example 75% of 300 is what?

part= (Percent/100) * whole x = (75/100) * 300

How to find y coordinate of vertex

plug in the # solved for the x coordinate in to the quadratic

How would you solve this problem What is the solution set to the equation? √2x+6 +4=x+3

plug in the possible answer choices in to the equation to see which one satisfies the equation

SOHCAHTOA

sine- opp/hyp cos- adj/hyp tan- opp/adj

Perpendicular lines have...

slopes that are negative reciprocals y=4x+3 y=-(1/4)+7

If 2a/b = 1/2 , what is the value of b/a?

solve the equation for b/a

x +1= 2/(x+1) In the equation above, which of the following is a possible value of x + 1 ?

solve the equation for x+1 1)Clear the fraction multiply both sides by x+1 (x+1)²=2 2) Take the square root of both sides x+1= √2 √2 is the answer

When given 2 equation or 2 points and 2 equations you can use...

substitution

if given two equations you can use

substitution or elimination

Average (mean)

sum of terms/ # of terms

When dealing with similar right triangles the sine of an angle of one triangle equals

the cosine of an angle of the other triangle

In a right triangle the sine of one acute angle equals

the cosine of the other acute angle

Range

the differnce between the lowest and highest terms in the set

parallel lines have...

the same slope Ex: y=5x+3 and y=5x+13

When making proportions

the same units equal each other

if a system of linear equations has no solution then...

the slopes of the equations have to be the same(lines are parallel) *same slope means graphs of the lines are parallel

in quadratic equation (ax²+bx+c) the c represents

the y-intercept (so the initial height)

In a triangle if two sides are equal then...

their angles must be congruent

if 24x² + 25x − 47/ ax-2 = 8x-3 -53/ax-2

then (8x-3)(ax-2)- 53 = 24x² + 25x − 47 the remainder goes over the outside term

if f(a)=0 if f(4)=0 x f(x) 0 3 2 1 4 0

then (x-a) or (x-4) is a factor of f(x)

If f(x) = 0.5 · x

then y or f(x) is directly proportional to x

If f(x) = 5/x

then y or f(x) is inversely proportional to x

a(x-h)²+k the bigger the a in vertex form the the smaller the a in vertex form

thinner the quadratic the wider the quadratic

How would you add 1/2x+1 + 5

to add 2 fraction they have to have the denominator 1) Multiply 5 by 2x+1/2x+1 10x+5/2x+1 2)Add both fractions together 10x+6/2x+1

average speed

total distance/ total time

GCF (Greatest Common Factor) Ex GCF (200,60)

use factor tree to find prime factors 200 = 2 × 2 × 2 × 5 × 5 60 = 2 × 2 × 3 × 5 multiply common prime factors 2 × 2 × 5 = 20

LCM(Least Common Multiple) Ex LCM (200, 60)

use factor tree to find prime factors 200 = 2 × 2 × 2 × 5 × 5 60 = 2 × 2 × 3 × 5 Multiply common prime factors the greatest # of times they appear in either # (every prime factor is included) 2 × 2 × 2 × 3 × 5 × 5

Probability Multliplication

use multiplication when finding probability of 2 or more independent events

mode

value in the list that appears most often

If x^a2/ x^b2=x¹⁶ , x > 1, and a+b= 2, what is the value of a-b?

when the bases are the same the exponents can be equal so 1) Rewrite x^a2/ x^b2=x¹⁶ as x^a2/ x^b2=16 Dividing exponents with the same base is the same as subtracting exponents 2)x^a2-b2=16 3)a2-b2=16 4) take the square root of both sides a-b=4

When is a rational expression undefined Ex: -3/x²+3x-10

when the denominater is = 0 x²+3x-10=0 (x+5)(x-2)=0 -5, 2

To solve for the y-intercept

x has to be 0

if (x+6)(x-0.7)=0 then

x+6= 0 x-0.7=0

Special Right Triangle (Isoceles Triangle)

x-x-x√2 the x's are across the 45° angles the x√2 is across the 90°

30-60-90 triangle

x-x√3-2x side opposite 30° is x side oppiste 60° is x√3 side opposite 90° is 2x

How to find x-coordinte of vertex

x=-b/2a

FOIL (x+a)(x+b)

x² +(b+a)x +ab

Exponent rule (multiplication) x³ * x²

x³ * x² = x³⁺² = x⁵

Exponent Zero Rule

x⁰ = 1 2⁰ = 1 2x⁰ = 2

Exponent rule (division) x⁴/x²

x⁴/x² = x⁴⁻² = x²

To solve for the x-intercept

y has to be 0

Exponential Growth

y= a(1+r)× a= initial value r= rate of growth x=time

Exponential Decay

y=a(1-r)× a= initial value r= rate of decay x=time

Vertex Form of a parabola/ quadratic

y=a(x-h)² +k h,k =vertex of a parabola

Slope intercept form

y=mx+b m=slope b= y-intercept

Absolute Value |5| |-5| Absolute Value is always_____ except_______

|5|= 5 |-5|= 5 Absolute Value is always_postive ____ except __if there is a negative in front of it except negative in front of it___

|x-4|= 3 is both

|x-4|= 3 |x-4|= -3

Area of a Circle

πr²

Area of a sector

πr²(central angle/360)

Volume of a Cylinder

πr²h

How to figure distance from point A to point B (Distance Formula)

√(x₂-x₁)² +(y₂-y₁)²

Radical Rule √xy

√xy = √x * √y