BYTETOOLS

Quadratic Equation Solver

Solve any quadratic equation ax² + bx + c = 0. Get real or complex roots, the discriminant, vertex and axis of symmetry with clear step-by-step working.

ax² + bx + c = 0

1
Discriminant
2
Root x₁
1
Root x₂
(1.5, -0.25)
Vertex

Two real roots

Axis of symmetry: x = 1.5

Discriminant Δ = b² − 4ac = -3² − 4·1·2 = 1

√Δ = 1

x = (−b ± √Δ) / (2a) = (3 ± 1) / 2

x₁ = 2, x₂ = 1

  • Solves any quadratic ax² + bx + c = 0
  • Real and complex roots handled
  • Discriminant shown to classify the roots
  • Vertex and axis of symmetry included
  • Step-by-step quadratic formula working
  • 100% private — runs entirely in your browser

How to use the Quadratic Equation Solver

  1. 1

    Enter the coefficient a (the x² term).

  2. 2

    Enter the coefficient b (the x term) and c (the constant).

  3. 3

    Read the discriminant and whether the roots are real or complex.

  4. 4

    Review the roots, vertex and axis of symmetry with the steps shown.

  5. 5

    Click Copy to grab the solution.

About the Quadratic Equation Solver

The ByteTools Quadratic Equation Solver solves ax² + bx + c = 0 for any coefficients you enter. It computes the discriminant to determine whether the roots are real or complex, then returns the roots, the vertex of the parabola and its axis of symmetry, with every step shown.

It is built for algebra and calculus students, teachers and engineers who need reliable roots and the working behind them. When the discriminant is negative, the solver returns the roots in exact complex form rather than simply saying there is no solution.

All the maths runs in your browser with JavaScript, so nothing is uploaded and results are instant and private. Copy the roots or the full solution with one click.

Frequently asked questions

What is the quadratic formula?

For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) ÷ (2a). The expression under the square root, b² − 4ac, is the discriminant and determines the nature of the roots. The solver applies this formula and shows each substitution.

What does the discriminant tell you?

The discriminant b² − 4ac reveals the roots without solving fully. If it is positive there are two distinct real roots, if it is zero there is one repeated real root, and if it is negative there are two complex conjugate roots.

How do you find the vertex of a parabola?

The vertex lies at x = −b ÷ (2a), and you get the y-coordinate by substituting that x back into the equation. This point is the minimum if a is positive or the maximum if a is negative, and the axis of symmetry passes through it.

What happens when the discriminant is negative?

There are no real roots, but there are two complex conjugate roots. The solver expresses them in the form p ± qi, using the square root of the absolute discriminant, instead of just reporting no solution.

What if a is zero?

If a is zero the equation is linear, not quadratic, so the quadratic formula does not apply. The solver detects this and solves the linear equation bx + c = 0 instead, or flags it when b is also zero.

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