Algebra Reference Sheet

Practice Algebra

Solving Linear Equations

One-Step Equations

x + a = b → x = b - a
x - a = b → x = b + a
ax = b → x = b/a
x/a = b → x = ab

Do the inverse operation to both sides.

Two-Step Equations

1. Undo addition/subtraction first
2. Then undo multiplication/division
Example: 2x + 5 = 13
2x = 8 → x = 4

Multi-Step Equations

1. Distribute if needed
2. Combine like terms on each side
3. Move variable terms to one side
4. Move constants to the other side
5. Divide to isolate the variable

Key Principle

Whatever you do to one side, you must do to the other side. The goal is always to isolate the variable.

Inequalities

Symbols

< less than
> greater than
≤ less than or equal to
≥ greater than or equal to

Critical Rule

When you multiply or divide both sides by a negative number, flip the inequality sign!

Example: -2x > 8 → x < -4

Systems of Equations

Substitution Method

1. Solve one equation for one variable
2. Substitute into the other equation
3. Solve for the remaining variable
4. Back-substitute to find the other

Elimination Method

1. Align like terms vertically
2. Multiply equations if needed so one variable cancels
3. Add or subtract equations
4. Solve for the remaining variable

Polynomials

Adding & Subtracting

Combine like terms (same variable and exponent)
When subtracting, distribute the negative sign
(3x² + 2x) - (x² - 4x) = 2x² + 6x

Multiplying (FOIL)

First: Multiply first terms
Outer: Multiply outer terms
Inner: Multiply inner terms
Last: Multiply last terms
(x + 2)(x + 3) = x² + 3x + 2x + 6 = x² + 5x + 6

Special Products

Perfect Square: (a + b)² = a² + 2ab + b²
Perfect Square: (a - b)² = a² - 2ab + b²
Difference of Squares: (a + b)(a - b) = a² - b²

Factoring

GCF (Greatest Common Factor)

Find the largest factor common to all terms
6x² + 12x = 6x(x + 2)
Always try GCF first!

Factoring Trinomials

x² + bx + c: find two numbers that
multiply to c AND add to b
x² + 5x + 6 = (x + 2)(x + 3)
because 2 × 3 = 6 and 2 + 3 = 5

Difference of Squares

a² - b² = (a + b)(a - b)
x² - 9 = (x + 3)(x - 3)
4x² - 25 = (2x + 5)(2x - 5)

Perfect Square Trinomials

a² + 2ab + b² = (a + b)²
a² - 2ab + b² = (a - b)²
x² + 10x + 25 = (x + 5)²

Quadratic Formula

For ax² + bx + c = 0:

x = (-b ± √(b² - 4ac)) / 2a

b² - 4ac > 0

Two real solutions

b² - 4ac = 0

One real solution

b² - 4ac < 0

No real solutions

Exponent Rules

Product Rule

x^a · x^b = x^a+b

Quotient Rule

x^a / x^b = x^a-b

Power Rule

(x^a)^b = x^ab

Zero Exponent

x⁰ = 1 (x ≠ 0)

Negative Exponent

x^-a = 1/x^a

Power of a Product

(xy)^a = x^ay^a

Rational Expressions

Simplifying

1. Factor both numerator and denominator
2. Cancel common factors
3. State excluded values (denominator ≠ 0)

Examples

(x² - 4) / (x + 2) = x - 2
(x² + 3x) / x = x + 3
Excluded: values that make denominator = 0

Loading Via Latina...