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Mathematics · Algebra

Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0 with the quadratic formula: both roots (real or complex), the discriminant and what it tells you, the vertex and axis of symmetry, the factored form, the sum and product of the roots, and a graph of the parabola — with every step shown.

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Two Real Roots

x = -1 or x = 3

2x² − 4x − 6 = 0

Discriminant (b² − 4ac)
64 — positive: two real roots
Vertex
(1, -8) — the minimum point
Axis of Symmetry
x = 1
y-intercept
(0, -6)
Factored Form
2(x + 1)(x − 3)
Sum and Product of Roots
sum −b/a = 2 · product c/a = -3

Quadratic formula, step by step

Solve 2x² − 4x − 6 = 0 with x = (−b ± √(b² − 4ac)) ÷ 2a

  1. 1: Step 1 · Discriminant

    D = b² − 4ac = (-4)² − 4 × 2 × -6

    Expression becomes 64

  2. 2: Step 2 · Square root

    √D = √64

    Expression becomes 8

  3. 3: Step 3 · Roots

    x = (4 ± √D) ÷ 4

    Expression becomes -1 or x = 3

Answer: x = -1 or x = 3

y = 2x² − 4x − 6

How this is calculated

x = (−b ± √(b² − 4ac)) ÷ 2a · Vertex = (−b/2a, c − b²/4a)

D = b² − 4ac
Discriminant: > 0 two real roots, = 0 one repeated root, < 0 two complex conjugate roots
Vieta
Sum of roots = −b/a, product = c/a

Every quadratic equation can be solved with the quadratic formula. The discriminant decides the kind of roots: positive gives two places where the parabola crosses the x-axis, zero means it just touches, negative means it never crosses and the roots are complex. To stay accurate when b is large, the roots are computed with the numerically stable form of the formula.

Assumptions & sources

  • Real coefficients; complex roots are shown as p ± qi.
  1. 1Mathematics, Class 10 — Quadratic Equations — NCERT
  2. 2Numerical Recipes — quadratic and cubic equations (stable root formula) — Press, Teukolsky, Vetterling & Flannery, Cambridge University Press

Calculator ID MATH-QUAD-001 · version 1.0.0

Common questions

What does the discriminant tell you?

b² − 4ac > 0: two different real roots; = 0: one repeated root; < 0: no real roots, two complex ones.

How do I solve 2x² − 4x − 6 = 0?

D = 16 + 48 = 64, √D = 8, so x = (4 ± 8) ÷ 4: x = 3 or x = −1.