The Quadratic Formula
Simple Explanation
The quadratic formula, x = (−b ± √(b² − 4ac)) / 2a, solves any equation of the form ax² + bx + c = 0 directly from its coefficients — no factoring required, and it always works, even when factoring is difficult or impossible.
Why Do We Need It?
Not every quadratic factors nicely — the quadratic formula is the universal method that solves every quadratic equation, guaranteed, using only the coefficients.
Formula
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Solves any quadratic equation ax² + bx + c = 0 directly from its coefficients, without needing to factor.
- x
- — the solution(s) — the roots of the equation
- a, b, c
- — the coefficients of ax² + bx + c = 0 (a ≠ 0)
- ±
- — produces two solutions — one using +, one using −, unless the discriminant is 0
When to use it: Whenever a quadratic equation cannot be factored easily (or at all), or you want a reliable method that always works.
Worked Example
Solve using the quadratic formula
Solve 2x² + 3x − 2 = 0.
Why Does This Work?
Completing the square on the general equation ax² + bx + c = 0 — dividing by a, moving c over, adding (b/2a)² to both sides to form a perfect square, and then taking the square root of both sides — algebraically derives exactly this formula, so it is guaranteed to solve every quadratic equation.
Real-Life Example
Finding when a ball lands
A ball's height is h(t) = −5t² + 20t + 1, and you need to find when it hits the ground (h = 0).
Since −5t² + 20t + 1 = 0 does not factor nicely, the quadratic formula is the direct, reliable way to solve for the exact landing time t.
Practice
Solve x² − x − 6 = 0 for the larger root.
MediumCommon mistake
Forgetting to divide the entire numerator (including the ± term) by 2a — the whole expression −b ± √(b²−4ac) must be divided by 2a, not just the −b part.
Quick Review
- x = (−b ± √(b² − 4ac)) / 2a solves any ax² + bx + c = 0.
- The ± gives two solutions (unless D = 0, giving one repeated solution).
- Derived from completing the square on the general quadratic equation.