How to Solve Algebra Equations
Algebra is the process of finding an unknown value by rearranging an equation using inverse operations. Linear equations require isolating the variable through addition, subtraction, multiplication, and division. Quadratic equations introduce an x² term and yield two possible solutions via the quadratic formula or factoring. This guide walks through both types with full worked examples.
Last updated: March 31, 2026
The Formula
Linear: ax + b = c → x = (c − b) / a Quadratic: ax² + bx + c = 0 → x = (−b ± √(b² − 4ac)) / 2a Discriminant: D = b² − 4ac
Variable Definitions
| Symbol | Name | Description |
|---|---|---|
| a, b, c | Coefficients | The numerical constants in the equation. In ax² + bx + c = 0, 'a' must not be zero. |
| x | Unknown | The value being solved for |
| D | Discriminant | b² − 4ac — tells you how many real solutions exist before doing the full calculation |
Step-by-Step Example
Solve: 2x² − 5x − 3 = 0
Given
Solution
- 1Identify coefficients:
a = 2, b = −5, c = −3 - 2Calculate discriminant:
D = (−5)² − 4(2)(−3) = 25 + 24 = 49 - 3Apply quadratic formula:
x = (5 ± √49) / 4 = (5 ± 7) / 4 - 4Solution 1:
x = (5 + 7) / 4 = 3 - 5Solution 2:
x = (5 − 7) / 4 = −0.5
x = 3 or x = −0.5. Verify: 2(9) − 15 − 3 = 0 ✓ and 2(0.25) + 2.5 − 3 = 0 ✓
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Common Mistakes to Avoid
Forgetting to move all terms to one side — the equation must equal zero before applying the quadratic formula.
Sign errors with b — when b is negative, −b becomes positive. Write it explicitly.
Dividing only part of the numerator by 2a — the entire (−b ± √D) is divided by 2a.
Stopping after one solution — quadratics always have two solutions (they may be equal or complex).