Quadratic Equation Solver
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula. You'll get both roots (real or complex), the discriminant that determines their nature, and the parabola's vertex.
How to use this calculator
- Write your equation in the form ax² + bx + c = 0 (move everything to one side).
- Enter a, b and c — including their signs.
- Press Solve to get the roots, discriminant and vertex.
- If a term is missing, its coefficient is 0 (x² − 9 = 0 means a=1, b=0, c=−9).
Formula used
The discriminant D = b² − 4ac decides everything: D > 0 gives two real roots, D = 0 one repeated root, D < 0 two complex roots. The vertex sits at x = −b/2a — always midway between the roots.
Example calculation
x² − 5x + 6 = 0 (a=1, b=−5, c=6):
D = 25 − 24 = 1 → x = (5 ± 1)/2 → x = 3 or x = 2. Check: sum of roots 5 = −b/a ✓, product 6 = c/a ✓ — and indeed (x−2)(x−3) = x² − 5x + 6.
What quadratic equations describe
Quadratics model anything with squared behavior: projectile motion (height vs time), areas, profit optimization, and every parabola in physics and engineering. Solving one means finding where the parabola crosses zero.
The quadratic formula always works, but the discriminant tells you the story before you finish: whether the parabola crosses the x-axis twice, touches it once, or floats entirely above or below it (complex roots). The sum (−b/a) and product (c/a) of the roots — Vieta's formulas — give a fast way to verify any answer.
Why use this calculator?
- Solve any quadratic — including ones that don't factor nicely or have complex roots.
- Verify homework via the built-in sum/product checks (Vieta's formulas).
- Get the vertex too, which most solvers skip but every graphing question needs.
Frequently asked questions
What does the discriminant tell me?
b² − 4ac determines the roots' nature without solving: positive means two distinct real roots, zero means one repeated root, negative means two complex conjugate roots. Its sign is the first thing to check in any exam question.
How do I solve a quadratic without the formula?
Try factoring first: find two numbers that multiply to a·c and add to b. For x² − 5x + 6, the numbers −2 and −3 work, giving (x−2)(x−3) = 0. If nothing factors cleanly, use the formula or complete the square.
What are complex roots and are they "real" answers?
When D < 0, the roots involve i = √−1 and come as a conjugate pair like 2 ± 3i. They're perfectly valid mathematically — they mean the parabola never touches the x-axis — and they matter in electronics, control systems and signal processing.