About the Quadratic Solver
What is the Quadratic Solver?
A quadratic equation has the form ax² + bx + c = 0 with a ≠ 0. Its graph is a parabola, and its solutions (roots) are the x-values where the parabola crosses the x-axis. Quadratics describe projectile motion, area problems, profit maximisation and many physics formulas.
Enter the three coefficients and the solver applies the quadratic formula. The discriminant b² − 4ac tells you what kind of roots to expect: positive gives two real roots, zero gives one repeated root and negative gives a pair of complex conjugate roots. The vertex is the turning point of the parabola.
How to use the Quadratic Solver
- a (coefficient of x²) — cannot be 0 (otherwise the equation is linear). (sample: 1)
- b (coefficient of x) — enter 0 if there is no x term. (sample: -5)
- c (constant) — enter 0 if there is no constant term. (sample: 6)
- Press Solve Equation (results also refresh as you type). The result card shows the Roots, along with Discriminant (b² − 4ac), Nature of roots, Vertex (h, k) and Sum of roots (−b/a).
Formula used
x = (−b ± √(b² − 4ac)) ÷ 2a
- a, b, c = the coefficients of x², x and the constant term
- D = the discriminant, b² − 4ac
Vertex: h = −b ÷ 2a and k = ah² + bh + c. By Vieta's formulas the two roots always add up to −b/a and multiply to c/a.
Worked example
These are the sample values already filled in the calculator above, and the result it returns for them:
| Input | Sample value |
|---|---|
| a (coefficient of x²) | 1 |
| b (coefficient of x) | -5 |
| c (constant) | 6 |
| Result | Value |
|---|---|
| Roots | x₁ = 3, x₂ = 2 |
| Discriminant (b² − 4ac) | 1 |
| Nature of roots | Two distinct real roots |
| Vertex (h, k) | (2.5, -0.25) |
| Sum of roots (−b/a) | 5 |
| Product of roots (c/a) | 6 |
Tips for using the Quadratic Solver
- Move every term to one side first: x² = 5x − 6 becomes x² − 5x + 6 = 0, so a = 1, b = −5 and c = 6.
- If a is negative the parabola opens downward and the vertex is a maximum — useful for maximum height or maximum profit problems.
- Check a root by substituting it back: for x = 2, 2² − 5 × 2 + 6 = 0.