Solve ax² + bx + c = 0 for real and complex roots — with discriminant analysis, factored form, vertex, axis of symmetry, and a live parabola plot.
Enter coefficients a, b, c — results and parabola update live.
Roots of ax² + bx + c = 0
Discriminant Δ
1
b² − 4ac
Sum x₁ + x₂
5
= −b/a
Product x₁·x₂
6
= c/a
Direction
∪ (opens up)
sign of a
Vertex (h, k)
(2.5, −0.25)
min/max point
Axis
x = 2.5
x = −b/(2a)
y-intercept
6
at x = 0
Factored form
(x − 3)(x − 2)
a(x − x₁)(x − x₂)
| Discriminant | Form | Geometric meaning |
|---|---|---|
| Δ > 0 | 2 real roots | parabola crosses x-axis at two points |
| Δ = 0 | 1 repeated real root | parabola touches x-axis at one point |
| Δ < 0 | 2 complex roots | parabola does not cross x-axis |
| Δ = perfect square | rational roots | factors cleanly over the integers |
The Method
Every quadratic ax² + bx + c = 0 can be solved by completing the square: rewrite as a(x + b/2a)² = (b² − 4ac) / 4a, take square roots, and rearrange. The result is the famous quadratic formula, x = (−b ± √Δ) / 2a, where Δ = b² − 4ac is the discriminant. Its sign tells you everything about the roots: positive gives two real roots, zero gives one repeated root, negative gives a complex conjugate pair. Even when there are no real roots, the parabola still has a clean vertex at (−b/2a, c − b²/4a).
Working for current coefficients
About This Tool
A quadratic formula calculator solves any equation of the form ax² + bx + c = 0 using the famous formula x = (−b ± √(b² − 4ac)) / 2a. Enter three coefficients and the calculator returns both roots (real or complex), the discriminant Δ = b² − 4ac, the vertex (h, k), the axis of symmetry, the y-intercept, and a clean factored form when one exists.
Quadratics show up everywhere in physics, engineering, finance, and computer graphics. Projectile motion uses them to find launch time and range. Optics uses them in lens equations. Finance uses them in compound-interest break-even problems. Anywhere a system depends on a squared term — area, energy (½mv²), gravity, parabolic reflectors — quadratics are the right tool.
The discriminant Δ tells you everything about the roots without solving: Δ > 0 means two distinct real roots, Δ = 0 means one repeated real root (a perfect square), and Δ < 0 means two complex conjugate roots — the parabola never crosses the x-axis. The calculator highlights which case applies and renders the parabola so you can see why.
Use this free quadratic formula calculator for homework, exam revision, projectile problems, optimisation, or any time you need a quick, accurate solution with full step-by-step working. All calculation is local — no sign-up, no tracking.
Real & Complex Roots
Handles every case — two real, one repeated, or two complex conjugate roots.
Live Parabola
SVG plot marks vertex, axis of symmetry, and real roots automatically.
Step-by-Step Working
Each substitution shown — ideal for homework write-up.
Vertex & Axis
Vertex (h, k), axis of symmetry x = −b/(2a), and y-intercept reported.
100% Free & Private
No account, no tracking — every calculation runs locally in your browser.
Factored Form
Renders a(x − x₁)(x − x₂) when roots are real.
From coefficients to full analysis in seconds.
Type the three coefficients into the equation display. Negative values are fine — just type the minus sign.
The headline shows x₁ and x₂. Complex roots are written in a ± bi form.
The Δ stat immediately tells you which root case applies — positive, zero, or negative.
The plot shows the curve, real roots (green dots) and vertex (orange dot) — useful for visualising the geometry.
Vertex, axis of symmetry, y-intercept, sum and product of roots, and factored form are all shown.
The formula card walks through Δ → √Δ → roots with substituted values — ideal for assignment write-up.
Everything you need to know about the quadratic formula and parabolas.
For any equation of the form ax² + bx + c = 0 with a ≠ 0, the roots are x = (−b ± √(b² − 4ac)) / (2a). The ± gives two solutions, and the expression inside the square root is the discriminant.
The discriminant Δ = b² − 4ac tells you the nature of the roots without solving. Δ > 0: two distinct real roots. Δ = 0: one repeated real root. Δ < 0: two complex conjugate roots — the parabola never crosses the x-axis.
The vertex of y = ax² + bx + c is the parabola's turning point. Its coordinates are (h, k) = (−b/(2a), c − b²/(4a)). It is the minimum if a > 0 and the maximum if a < 0.
The axis of symmetry is the vertical line through the vertex, x = −b/(2a). The parabola is a mirror image about this line, so for any value of x, the y-value equals the y-value at the mirrored point.
Vieta's formulas relate roots to coefficients: x₁ + x₂ = −b/a and x₁·x₂ = c/a. These give a quick sanity check on any answer — if your sum and product don't match, your roots are wrong.
If a = 0 the equation is no longer quadratic — it's linear: bx + c = 0 with single root x = −c/b. The calculator flags this case and offers the linear solution.
Yes — when Δ < 0, the calculator reports the two complex conjugate roots as x = α ± βi where α = −b/(2a) and β = √(−Δ)/(2a). These solve the equation in the complex number system.
When the discriminant is a perfect square (and a, b, c are integers), the roots are rational and the polynomial factors cleanly over the integers. E.g. x² − 5x + 6 has Δ = 1 (= 1²) and factors as (x − 2)(x − 3).
Everywhere with a squared term: projectile motion (h = v₀t − ½gt²), area-cost trade-offs, parabolic reflectors (satellite dishes), kinetic energy (KE = ½mv²), break-even points in finance, and the optimisation of one-variable cost / revenue functions.