Quadratic Equation Solver
Quick Examples
Step-by-Step Solution
Free Quadratic Equation Solver with Steps
Our quadratic equation solver uses the quadratic formula to find the roots of any equation in the form ax² + bx + c = 0. Enter the values of a, b and c and get instant results with a full step-by-step solution showing how the answer was reached — perfect for students learning algebra, verifying homework answers or checking exam solutions.
Discriminant and Nature of Roots
| Discriminant (D = b²−4ac) | Nature of Roots | Example |
|---|---|---|
| D > 0 | Two distinct real roots | x²−5x+6=0 → x=2, x=3 |
| D = 0 | One repeated real root | x²−4x+4=0 → x=2 (twice) |
| D < 0 | Two complex (imaginary) roots | x²+2x+5=0 → x=−1±2i |
Frequently Asked Questions
How do you solve a quadratic equation using the formula?
Write the equation in standard form ax²+bx+c=0. Calculate D=b²−4ac. Then x = (−b ± √D) / 2a. If D is positive, you get two real roots. If zero, one repeated root. If negative, two complex roots.
What does it mean when the discriminant is negative?
A negative discriminant (D < 0) means the quadratic equation has no real solutions. The roots are complex numbers involving the imaginary unit i (where i² = −1). These roots always appear as a conjugate pair: p+qi and p−qi.
Can a quadratic equation have only one root?
Yes. When D = 0, the equation has one repeated real root. This is called a perfect square trinomial. For example, x²−4x+4=0 factors as (x−2)² = 0, giving x = 2 as the only root.
Is this quadratic solver free?
Yes, 100% free. No account or registration needed. All calculations happen instantly in your browser.