OmniCalcsThe Quadratic Formula, Explained
Every quadratic equation, solved with one formula — and what the discriminant tells you.
Quadratic equations show up everywhere — from projectile motion in physics to profit maximization in business. The good news: one formula solves them all.
The General Form
A quadratic equation has the shape:
where a, b and c are numbers, and a ≠ 0. The solutions are given by:
The Discriminant Decides Everything
The part under the square root, b² − 4ac, is called the discriminant. It tells you what kind of solutions to expect:
- Positive → two distinct real roots.
- Zero → exactly one repeated real root.
- Negative → two complex roots (no real crossing of the x-axis).
Worked Example
Solve x² − 5x + 6 = 0 (a = 1, b = −5, c = 6):
Roots: (5 ± 1) ÷ 2 → x = 3 and x = 2
Check: 3² − 5(3) + 6 = 9 − 15 + 6 = 0 ✓ and 2² − 5(2) + 6 = 4 − 10 + 6 = 0 ✓.
The Vertex (the Turning Point)
The parabola's vertex sits at:
Plug that x back in to get the y-coordinate. It's a minimum when a > 0 and a maximum when a < 0 — handy for optimization problems.
Why It Matters in Real Life
Businesses use it to find the price that maximizes profit. Physicists use it to model the path of a ball. Anyone pricing a subscription, a flight or a loan is effectively working with a quadratic. Understanding the formula means understanding where that optimum sits — not just trusting a spreadsheet.
Skip the algebra when you're in a hurry with our free Quadratic Equation Solver, and refresh related skills with the Exponent Calculator.