Quadratic Equation Calculator
Solve ax² + bx + c = 0 online free. Enter coefficients a, b, c. Get the discriminant (b²−4ac), both roots (real or complex), and a full step-by-step quadratic-formula breakdown. 100% browser-based, no signup.
must be ≠ 0
- Identify coefficients: a = 1, b = -5, c = 6.
- Compute discriminant: Δ = b² − 4ac = (-5)² − 4·(1)·(6) = 1.
- Δ > 0 → two distinct real roots.
- Apply x = (−b ± √Δ) / 2a = (5 ± √1) / 2.
- x₁ = 3, x₂ = 2.
Δ > 0, so there are two distinct real roots.
Quadratic Formula
Solves ax² + bx + c = 0 with the quadratic formula x = (−b ± √(b² − 4ac)) / 2a. The calculator returns both roots and identifies whether they are real or complex.
Discriminant Insight
Shows Δ = b² − 4ac and explains what it means: Δ > 0 → two distinct real roots, Δ = 0 → one double root, Δ < 0 → two complex conjugate roots (a ± bi).
Step-by-Step
A full breakdown is shown: identify a/b/c, compute Δ, evaluate the formula at each ± branch. Great for checking algebra homework or verifying hand calculations.
100% Private
All the math runs in your browser. No coefficients are uploaded, no signup, and there is no daily limit. Use it offline on any device.
What is the quadratic formula?
For any equation of the form ax² + bx + c = 0 with a ≠ 0, the two roots are x = (−b ± √(b² − 4ac)) / 2a. The expression under the square root, b² − 4ac, is called the discriminant (Δ) and it tells you up front whether the roots are real and distinct, real and equal, or complex.
What does the discriminant tell me?
Δ = b² − 4ac. If Δ > 0 there are two distinct real roots. If Δ = 0 there is one real root of multiplicity two (a double root). If Δ < 0 there are no real roots — instead there are two complex conjugate roots of the form a ± bi.
How are complex roots displayed?
When the discriminant is negative the calculator writes each root in the form "real ± imag·i" using i = √(−1). For example, the roots of x² + 1 = 0 are shown as "i" and "−i", and the roots of x² − 2x + 5 = 0 are shown as "1 + 2i" and "1 − 2i".
What if a = 0?
When a = 0 the equation is not quadratic — it is linear (bx + c = 0) and has at most one real root, x = −c/b. The calculator detects this case and switches to linear mode automatically. If both a and b are zero, the equation is either an identity (always true when c = 0) or has no solution (when c ≠ 0).
Is the calculator free?
Yes — 100% free, no signup, and no daily limit. It runs entirely in your browser so you can use it offline on a phone, tablet, or laptop.
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