Solving linear and quadratic equations
ax + b = 0, the discriminant, the quadratic formula: how to solve a linear or quadratic equation, step by step.
Linear equation
ax + b = 0 (a ≠ 0) x = −b / a
Example: 3x + 6 = 0 → x = −6/3 = −2
Quadratic equation: the discriminant
ax² + bx + c = 0 (a ≠ 0) Δ = b² − 4ac
| Sign of Δ | Number of solutions |
|---|---|
| Δ > 0 | 2 distinct real solutions |
| Δ = 0 | 1 real solution (double root) |
| Δ < 0 | No real solution (2 complex solutions) |
The quadratic formula
x = (−b ± √Δ) / (2a)
A worked example
x² − 5x + 6 = 0 Δ = (−5)² − 4×1×6 = 25 − 24 = 1 x = (5 ± 1) / 2 → x = 3 or x = 2
Quick check: the sum of the two solutions always equals −b/a, and their product always equals c/a — for the example above, 3+2=5=−(−5)/1 ✓ and 3×2=6=6/1 ✓.
Factoring instead of computing the discriminant
When possible, factoring directly is faster: x² − 5x + 6 = (x−2)(x−3), so the solutions are obvious (2 and 3) without computing Δ.
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