Quadratic Equations Mastery Guide – Complete Notes, Step-by-Step Solutions & Practice Tests

INTRODUCTION TO ALGEBRA

Algebra is the study of numbers, variables, and relationships. It
helps solve real-life problems like budgeting, measurements, and
data analysis.
A variable represents an unknown value.
Example:
x + 5 = 12 → x is the unknown

IMPORTANT TERMS
Variable – symbol (x, y) representing a number
Constant – fixed number (e.g., 10)
Coefficient – number multiplied by a variable (e.g., 4x)
Expression – no equal sign (2x + 3)
Equation – has equal sign (2x + 3 = 7)

➕ ORDER OF OPERATIONS (PEMDAS)
Solve expressions in this order:
Parentheses
Exponents
Multiplication & Division
Addition & Subtraction

Example:
3 + 2 × 5 = 3 + 10 = 13
➗ SIMPLIFYING EXPRESSIONS

Combine like terms:

Example:
3x + 2x = 5x
4y + 3 + y = 5y + 3

Only combine terms with the SAME variable.
SOLVING LINEAR EQUATIONS

Steps:
1. Simplify both sides
2. Move variables to one side
3. Move constants to the other
4. Divide to isolate variable

Example:
2x + 4 = 12
2x = 8
x=4

, ➖ MULTI-STEP EQUATIONS
Example:
3x + 5 = 2x + 9
3x - 2x = 9 - 5
x=4

➗ EQUATIONS WITH PARENTHESES
Use distributive property:

Example:
2(x + 3) = 10
2x + 6 = 10
2x = 4
x=2
EQUATIONS WITH VARIABLES ON BOTH SIDES

Example:
5x + 2 = 3x + 10
5x - 3x = 10 - 2
2x = 8
x=4

➗ FRACTIONS IN EQUATIONS
Multiply both sides to remove fractions:

Example:
x/4 = 3
x = 12

INEQUALITIES
Use same steps as equations but remember:
Flip the sign when multiplying/dividing by a negative

Example:
-2x > 6

PRACTICE TEST
EASY
x + 6 = 10
2x = 14
x-5=9

MEDIUM
3x + 2 = 11
2(x + 4) = 12
4x - 3 = 13