The quadratic formula
Types of solutions
| Discriminant | Roots | Graph |
|---|---|---|
| D > 0 | Two distinct real roots | Crosses the x-axis twice |
| D = 0 | One repeated real root | Touches the x-axis at the vertex |
| D < 0 | Two complex roots p ± qi | Does not reach the x-axis |
Sum and product of roots (Vieta’s formulas): x₁ + x₂ = −b/a and x₁ × x₂ = c/a — a quick way to check an answer.
How to use the calculator
- Write your equation in standard form ax² + bx + c = 0 — move all terms to one side first.
- Enter a, b and c. Coefficient a must not be 0 (otherwise the equation is linear).
- Read the discriminant, the roots (real or complex), the vertex and the factored form, with each step shown.
Other ways to solve quadratics
Factoring is fastest when the roots are whole numbers. For x² − 5x + 6 = 0, find two numbers that multiply to 6 and add to −5: −2 and −3. So (x − 2)(x − 3) = 0 and x = 2 or x = 3.
Completing the square rewrites the equation as a(x − h)² + k = 0. For x² + 6x + 5 = 0: x² + 6x + 9 = 4, so (x + 3)² = 4, x + 3 = ±2, and x = −1 or x = −5. This method is how the quadratic formula itself is derived.
The quadratic formula always works, including when the roots are irrational or complex.
Vertex and graph
The graph of y = ax² + bx + c is a parabola. It opens upward if a > 0 and downward if a < 0. The vertex — the minimum or maximum point — is at x = −b ÷ 2a. The roots are where the parabola crosses the x-axis; if the discriminant is negative, it never crosses.
Useful checks
- Sum and product of roots (Vieta’s formulas): x₁ + x₂ = −b/a and x₁ × x₂ = c/a. For 2x² − 4x − 6: sum = 2 and product = −3, which matches 3 and −1.
- Substitute back: 2(3)² − 4(3) − 6 = 18 − 12 − 6 = 0 ✓.
Where quadratics appear
- Physics: the height of a thrown ball, h = −4.9t² + v₀t + h₀ (in meters), is quadratic in time.
- Geometry: finding dimensions from an area, such as a rectangle whose length is 3 more than its width and area is 40 (w² + 3w − 40 = 0, so w = 5).
- Business: profit as a function of price often forms a parabola, and the vertex gives the price that maximizes profit.