Algebra Summary Calculator

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Solving Equations and Inequalities (1 unknown variable)

1) Isolate variable on one side of the equation
2) Move all constants over to the other side of the equation
4) If there is a constant in front of the variable, divide each side of the equation by this constant to remove it

System of Equations (2 unknowns)

Elimination Method - The goal is to remove one variable by subtracting one equation from the other
ax + by = c
dx + ey = f
We need to find a number (n) to multiply top and bottom equations by such that na = nd or nb = ne so that we can eliminate the x terms or y terms

Substitution Method - Substitute one equation into the other
Rearrange Equation dx + ey = f, we get

x  =	f - ey
	d

We then substitute that into Equation 1 where we see the variable x
Cramers Rule - Learn the 3 shortcut equations
Δ = a * e - b * d
Variable 1 Numerator = c * e - b * f
Variable 2 Numerator = a * f - c * d

Variable 1  =	Variable 1 Numerator
	Δ

Variable 2  =	Variable 2 Numerator
	Δ

Slope and Line Equations

Given 2 points (x₁,y₁) and (x₂,y₂), the slope of the line (m) is:

m  =	y2 - y1
	x2 - x1

The line equation is y = mx + b

Parallel, Perpendicular, and Intersecting Lines

* 2 lines are parallel if their slopes equal each other
* 2 lines are perpendicular if their slopes are negative reciprocals of each other, i.e., m and -1/m.
* 2 lines are just intersecting if both conditions above fail.

Binomial Multiplication (FOIL)

(a + b)(c + d) is expanded using First-Outside-Inside-Last (FOIL)
ac + ad + bc + bd

Quadratic Equations

ax² + bx + c = 0
When a = 1, you solve these by finding two numbers that x and y such that x + y = b and xy = c

Rational Exponents

a^b/c = (^c√a)^b is solved by doing the following
Take the c^th root of b. Then, take that value, and raise it to the power of b
27^2/3 = ³√27 = 3 --> 3² = 9

Imaginary Numbers:

i = √-1
i²= -1
i³= -i
i⁴= 1
They key is to find the maximum power you can of 4 since i⁴ = 1

Difference of Two Squares:

ax² - by² = (ax + by)(ax - by)

Factoring Polynomials:

Sum of Cubes: a³ + b³ = (a + b)(a² - ab + b²)
Difference of Cubes: a³ - b³ = (a - b)(a² + ab + b²)
If there is not a shortcut formula, you first factor any constants that you can. Next, you go through each variable in the expression. If every term contains a variable, that variable can be factored out using the lowest power that you see in that expression, else, it cannot be factored.

Factoring Rational Expressions:

x2 + 6x + 8
x2 + 9x + 20

Factor the top and bottom to get this answer