Quadratic Equation Solver
Solve ax² + bx + c = 0 — real and complex roots
Coefficients
x₁ = , x₂ =
x =
± i
Quadratic Equation Solver — Complete Guide & Instructions
MakeCard Quadratic Equation Solver solves any equation of the form ax² + bx + c = 0 the moment you press Solve. Enter the three coefficients and the tool applies the quadratic formula, computes the discriminant, and tells you exactly what kind of roots you have — two distinct real roots, one repeated root, or a complex conjugate pair shown in a ± bi form. The equation is displayed live as you type so you can catch input mistakes early.
How to Use Quadratic Equation Solver
- 1 Enter the coefficients a, b and c into the three boxes.
- 2 Check the equation preview below the inputs to confirm it reads correctly.
- 3 Press “Solve”.
- 4 Read the result: two real roots, one repeated root, or complex roots — with the discriminant D shown.
- 5 Adjust any coefficient and solve again for a different equation.
Key Features & Capabilities
- Solves ax² + bx + c = 0 via the quadratic formula.
- Computes the discriminant D = b² − 4ac and reports the root type.
- Handles all three cases: two real roots, one repeated root, complex conjugate roots (a ± bi).
- Live equation preview as you type the coefficients.
- Clear labeling — x₁/x₂ for real roots, real ± imagi for complex.
- Instant, in-browser math — no page reloads.
Supported Formats & Specifications
| Parameter | Specification |
|---|---|
| Input | Coefficients a, b, c (a must be non-zero for a true quadratic) |
| Method | Quadratic formula with discriminant D = b² − 4ac |
| Root Types | Two real · one repeated (D = 0) · complex conjugate pair |
| Output | Roots + discriminant value |
| Processing | Client-side, instant |
Practical Everyday Use Cases
- Homework and exam prep for class 9–12 algebra.
- Finding when a projectile or profit function crosses zero.
- Factoring quadratics by first finding the roots.
- Checking manual quadratic-formula working.
- Solving physics equations that reduce to a quadratic (e.g. s = ut + ½at² for t).
Pro Tips for Best Results
- If a = 0 the equation is linear, not quadratic — solve bx + c = 0 directly.
- A negative discriminant means no real roots — the tool reports the complex pair instead.
- When roots come out messy decimals, they were likely irrational — the exact surd form is (−b ± √D) / 2a.
- Use the sign convention carefully: in x² − 5x + 6 = 0, b is −5, not 5.
Common Problems & Solutions
A true quadratic needs a ≠ 0; if a is 0 it is a linear equation with a single solution x = −c/b.
That means D < 0 — the parabola never crosses the x-axis. The a ± bi result is still mathematically correct.
Double-check the signs of b and c — a missing minus flips the whole answer.
Privacy & Security
100% Client-Side Execution
Calculations run entirely in your browser — coefficients and results are never sent anywhere.
Frequently Asked Questions
Everything you need to know about using Quadratic Equation Solver
x = (−b ± √(b² − 4ac)) / 2a — the tool applies this for you and shows the discriminant D = b² − 4ac.
D < 0 means no real roots — the two solutions are a complex conjugate pair, shown as a ± bi.
The equation has one repeated real root — the parabola just touches the x-axis at x = −b/2a.
A zero a makes it linear, not quadratic — the tool will flag it, since the formula divides by 2a.
If the roots are r₁ and r₂, the equation factors as a(x − r₁)(x − r₂) = 0.
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