Download ALGEBRA CHEAT SHEET

ALGEBRA CHEAT SHEET
LINEAR EQUATIONS
1)
Point Slope Formula
y – y1 = m ( x – x 1 )

use when you have one point and the slope of the line

if you have two known points, first find the slope and then apply this formula

can be used for putting ordered pairs into standard form

if you solve for y, then you can put into slope-intercept form (and then into standard form if you wish)
2)
Slope-intercept Formula
y=mx+b

use when you know the slope (m) and the y-intercept (b)

if the equation is already in this form, then use to find the slope and the y-intercept

use to draw a linear graph since you know one point and the slope
3)
Slope Formula
m = y 2 – y1 / x 2 - x 1

rise over run (change in y’s divided by change in x’s)

use when you need to find the slope of two points

after finding the slope you can pick one of the points to develop the point slope formula, etc.
4)
Standard Form
Ax + By = C

can be derived from the point slope formula or the slope intercept formula

when deriving from two points, it doesn’t matter which point is used, the same standard form will result

slope m = -A/B
y-intercept b = C/B
5)
Parallel and Perpendicular Lines

parallel lines have the same slope, perpendicular lines have negative reciprocal slopes
FACTORING – always factor first
1)
FOIL
(first, outside, inside, last)
2)
Factoring Trinomials
-in the form of: ax 2 + bx + c

multiply a and c

find two factors of the product of a and c that equal b (watch signs)

rewrite the middle term (b) as the sum of these two factors

To group (if needed):

group the first two terms and the last two terms

take out the GCF of each group

rewrite the answer as two grouped multiples

check
3)
Perfect Square Trinomials

(x + y)2 = x2 + 2xy + y2
(refer to square of a sum)

(x - y)2 = x2 - 2xy + y2
(refer to square of a difference)

these are NOT differences of squares

requirements:

first term is a perfect square

third term is a perfect square

middle term is 2 or –2 times the product of the square root of the first and the square root of the last terms.
4)
Polynomials

only combine LIKE TERMS

watch negative signs – distribute them appropriately
5)
Differences of Squares

x2 – y2 = (x – y)(x + y) = (x + y)(x – y)

if the terms of the binomial have a common factor, factor out the GCF first
6)
Special Products

square of a sum

square of a difference
 difference of squares
LAWS OF EXPONENTS
 X-a = 1/Xa
(x + y)2 = (x + y) (x + y) = x2 + 2xy + y2
(x - y)2 = (x - y) (x - y) = x2 - 2xy + y2
(x + y) (x - y) = (x - y) (x + y) = x2 - y2




(Xa) (Xb) = Xa+b
(Xa)b = Xab
Xa/Xb = Xa-b
X0 = 1 (remember, anything to the zero power is ONE)
EXPONENTIAL FUNCTIONS
1)
General information:

first factor the base so that you have like bases on each side of the equation

then compare exponents to solve for the unknown
QUADRATIC FUNCTIONS & EQUATIONS
1)
General information:

Quadratic equations equal “0” while quadratic functions equal “y”

have either 0, 1, or 2 roots

roots (the solutions) are the points where the graph crosses the x-axis. Can be found by:

graphing, or factoring and setting each factor equal to zero, or by using the quadratic formula

the standard form is ax2 + bx + c

resulting graph is a parabola with a minimum or a maximum

the vertex is a minimum if the “a” coefficient is positive

the vertex is a maximum if the “a” coefficient is negative
2)
Formulas and use:

axis of symmetry x = -b / 2a

use to find the x component of the vertex

then substitute into original equation to find the y component

defines the equation of the axis of symmetry


Note use of square roots –
they are shown using the
words square root.
quadratic equation
x = -b +/- square root of b2 – 4ac / 2a

use to find the roots to any quadratic equation

watch the signs of a, b, c

results will be undefined if the square root has a negative value under it
discriminate
b2 – 4ac

use to determine if a quadratic is factorable

if positive (2 solutions), if zero (one solution), if negative (no real solutions)
RATIONAL EXPRESSIONS & EQUATIONS
1)
To simplify:
 use prime factorization to pair up like terms
 identify excluded values before canceling (those that make the denominator equal to ZERO – meaning undefined)
 cancel out like terms
2)
To multiply:
 first factor where possible and then cancel where possible (be careful with exponents)
 multiple straight across (don’t leave anything out)
 reduce and simplify
3)
To divide:
 flip the second term and follow multiplication rules
4)
To add and subtract expressions:
 find a common denominator (LCD works best) by factoring each denominator
 change each expression into an equivalent one with the LCD by multiplying each term by a factor to produce the LCD
 add or subtract and simplify
5)
Solving equations:
 goal is to eliminate the denominators so identify the LCD and multiply each side of the EQUATION by the
LCD to clear out the denominators
 solve for the variable (solutions that are excluded values are not actual solutions)
 resistance: series RT = R1 + R2
parallel 1/RT = 1/R1 + 1/R2
RADICAL EXPRESSIONS & EQUATIONS
1)
Pythagorean Theorem
 c2 = a2 + b2 where a & b are legs and c is the hypotenuse (if true then the triangle is a right triangle)
2)
Radical Expressions
 square root of ab = squate root of a X square root of b
 square root of a/b = square root of a / square root of b
 like radicands – add or subtract the coefficients of the like radicands
 unlike radicands – simplify to like radicands if possible, then add or subtract the coefficients of the like radicands
 multiply polynomials by CONJUGATE (similar to difference of perfect squares)
3)
Radical Equations
 equations with variables in the radicand – isolate the radical with the variable to one side then square both sides
 distance formula – use to find distance between 2 points on the coordinate plane

d = square root of (x2 – x1)2 + (y2 – y1)2

can be used to determine if a triangle is isosceles (2 equal sides)
 completing the square – used to make a quadratic expression a perfect square

find ½ of b, square the result, add this result to the original expression
be sure to add the resulting number to BOTH sides of the EQUATION