Quadratic Equation Solver
Solve ax² + bx + c = 0 instantly using the quadratic formula.
What this calculator does
Solves any quadratic equation in the form ax² + bx + c = 0, returning both real roots (or complex roots, if applicable) instantly using the quadratic formula.
Who this is for
Students checking homework or studying for an algebra exam, anyone who needs to solve a quadratic as part of a larger problem (physics, engineering, finance), or teachers generating quick answer keys.
Methodology
Solves ax² + bx + c = 0 using the quadratic formula: x = (−b ± √(b² − 4ac)) ÷ 2a. The discriminant, b² − 4ac, determines how many real solutions exist before the square root is even taken.
Worked example
Solving x² − 5x + 6 = 0: here a=1, b=−5, c=6. Discriminant = (−5)² − 4(1)(6) = 25 − 24 = 1 (positive, so two real roots). x = (5 ± √1) ÷ 2 = (5 ± 1) ÷ 2, giving x = 3 and x = 2. You can verify: (x−2)(x−3) = x² − 5x + 6, which matches the original equation.
Interpretation
If the discriminant is positive, the equation has two distinct real roots — the parabola crosses the x-axis at two points. If it's exactly zero, there's one repeated real root — the parabola just touches the x-axis at its vertex. If it's negative, there are no real roots at all — the parabola never touches the x-axis, and the two solutions are complex numbers involving i (the imaginary unit, where i² = −1).
Common mistakes
- Forgetting the ± sign. Every quadratic with a positive discriminant has two solutions, not one — both the addition and subtraction case.
- Sign errors on b. If your equation is x² − 5x + 6 = 0, then b = −5, not 5 — a common source of wrong answers.
- Assuming a negative discriminant means "no solution" entirely. It means no real solutions — complex solutions still exist mathematically.
- Forgetting to check for a common factor first. An equation like 2x² − 10x + 12 = 0 solves correctly with the formula directly, but simplifying to x² − 5x + 6 = 0 first (dividing by 2) makes the arithmetic easier and less error-prone.
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Frequently Asked Questions
How do you know how many solutions a quadratic has before solving it?
Check the discriminant (b² − 4ac) first. Positive means two real solutions, zero means exactly one repeated real solution, and negative means no real solutions (only complex ones).
What if the discriminant is negative?
A negative discriminant means the equation has no real roots — only complex roots involving the imaginary unit i. The parabola never crosses the x-axis.
What if a equals zero?
If a is zero, the equation is no longer quadratic — it becomes linear (bx + c = 0) and has at most one solution, not two.
What does the discriminant tell you?
It determines the nature of the roots: positive means two distinct real roots, zero means one repeated real root, and negative means two complex roots.