Quadratic Formula Calculator
Solve ax² + bx + c = 0 for real and complex roots, with the discriminant, exact radical form, the parabola’s vertex, axis, focus and directrix, the factored and vertex forms, a graph, and every step of the working.
Written by Suhaib Hassan. The quadratic formula is standard algebra, and every worked example is re-derived by hand. Our methodology.
It computes the discriminant Δ = b² − 4ac, classifies the roots, and simplifies square roots of integers into exact radicals such as (−5 ± √13) ÷ 2.
Most solvers give a decimal and stop. This one shows why the roots are what they are: the discriminant, the parabola, and the factored form.
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Graph of the parabola
y = ax² + bx + c with its roots, vertex, axis of symmetry and y-intercept.
Step-by-step working
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What the discriminant tells you
The sign of Δ = b² − 4ac decides the nature of the roots.
| Discriminant | Roots | Graph | Example |
|---|---|---|---|
| Δ > 0, perfect square | Two distinct rational roots | Crosses the x-axis twice | x² − 5x + 6 = 0 → 2, 3 |
| Δ > 0, not a square | Two distinct irrational roots | Crosses the x-axis twice | x² − 2 = 0 → ±√2 |
| Δ = 0 | One repeated real root | Touches the x-axis at the vertex | x² − 6x + 9 = 0 → 3 |
| Δ < 0 | Two complex conjugate roots | Never meets the x-axis | x² + 2x + 5 = 0 → −1 ± 2i |
Worked examples
Projectile height
A ball is thrown up at 20 m/s from 1.5 m: h(t) = −4.9t² + 20t + 1.5. When does it land, and how high does it go?
landing: t = (20 + √429.4) ÷ 9.8 = 4.155 s
peak at t = 20 ÷ 9.8 = 2.041 s, height 21.908 m
Golden ratio
The golden ratio solves x² − x − 1 = 0.
x = (1 ± √5) ÷ 2
φ = 1.618034, other root −0.618034
Complex roots
x² + 2x + 5 = 0.
x = (−2 ± 4i) ÷ 2
x = −1 ± 2i
What is the quadratic formula?
Any equation ax² + bx + c = 0 with a ≠ 0 has solutions x = (−b ± √(b² − 4ac)) ÷ 2a. It comes from completing the square on the general equation. The expression under the root, Δ = b² − 4ac, is the discriminant.
The parabola behind the equation
- Vertex: (h, k) with h = −b ÷ 2a and k = c − b² ÷ 4a. It is the minimum if a > 0 and the maximum if a < 0.
- Axis of symmetry: x = h; the roots sit symmetrically on either side of it.
- Focus and directrix: (h, k + 1 ÷ 4a) and y = k − 1 ÷ 4a.
- Forms: standard ax² + bx + c; factored a(x − x₁)(x − x₂); vertex a(x − h)² + k.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- NIST Digital Library of Mathematical Functions, section 1.11 (Zeros of polynomials) and chapter 1, Algebraic and Analytic Methods.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
Frequently asked questions
What happens if a = 0?
The equation is no longer quadratic: it becomes bx + c = 0, a linear equation with the single solution x = −c ÷ b (if b ≠ 0). The quadratic formula divides by 2a, so it cannot be used. This calculator tells you the linear solution instead.
How do complex roots appear on a graph?
When Δ < 0 the parabola does not touch the x-axis, so there are no real roots. The complex roots p ± qi have the real part p = −b ÷ 2a, which is the axis of symmetry, and the imaginary part q = √(−Δ) ÷ 2a, which measures how far the vertex is from the axis.
Why is the vertex’s x-coordinate the midpoint of the roots?
The two roots are −b ÷ 2a ± √Δ ÷ 2a, so their average is −b ÷ 2a — exactly the vertex’s x-coordinate. A parabola is symmetric about its axis.
What is the sum and product of the roots?
By Vieta’s formulas, x₁ + x₂ = −b ÷ a and x₁ · x₂ = c ÷ a. They let you check your answers: for x² − 5x + 6 the roots 2 and 3 add to 5 and multiply to 6.
When should I use factoring or completing the square instead?
Factoring is quickest when the roots are integers or simple fractions. Completing the square derives the formula and gives the vertex form. The quadratic formula always works, whatever the coefficients.
Can the coefficients be decimals or fractions?
Yes. Enter decimals directly. Exact radical forms are shown when the coefficients are integers; otherwise you get precise decimal roots.
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