Pre-Calculus 11 · Mathematics · Lesson 4.3
Quadratic formula
- Level
- Grade 11
- Reading time
- 6 min reading
- Review
- Last reviewed 2026-09-03
- Standard
- Accuracy standard
- Errata
Abstract
The general solution.
What is the quadratic formula?
Short answer
The quadratic formula solves any quadratic equation ax² + bx + c = 0 for x: x = (−b ± √(b² − 4ac)) / (2a). It works even when the quadratic cannot be factored easily. The expression under the root, b² − 4ac, is the discriminant, which tells you how many real solutions the equation has.
Concept preview
What you will be able to do
- 01Write any quadratic in standard form and read off , , with their signs.
- 02Solve any quadratic using .
- 03Simplify radical answers and give exact or rounded roots as required.
- 04Use the discriminant to predict how many real roots exist.
- 05Apply the formula to problems where factoring is impossible.
Factoring is wonderfully fast — when it works. But it only works when the roots happen to be nice fractions, and most quadratics are not that generous. Take and hunt for two integers that multiply to and add to : there is no such pair. The equation still has two perfectly good real roots — they are just irrational, tangled up in a square root. When factoring stalls like this, you need a tool that works on every quadratic without exception: the quadratic formula.
School Plan
Quadratic formula
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Related topics
Frequently asked questions
- What does the discriminant tell you?
- If b² − 4ac > 0 there are two real roots; if it equals 0 there is one; if it is negative there are no real roots.
- When should you use the quadratic formula instead of factoring?
- When the quadratic does not factor with simple integers — the formula always works.