PFVPFV Tools
All tools

Quadratic Equation Solver

Solve ax² + bx + c = 0 — real or complex roots, vertex, axis of symmetry.

  • quadratic
  • roots
  • discriminant
  • vertex
  • parabola

About Quadratic Equation Solver

A quadratic equation has the form ax² + bx + c = 0, and its roots — the values of x that make the equation true — are given by the quadratic formula. The discriminant (b² − 4ac) determines how many real roots there are: positive means two distinct real roots, zero means one repeated real root, negative means a conjugate pair of complex roots.

Beyond the roots, every quadratic graphs as a parabola with a vertex, an axis of symmetry, and a y-intercept that summarise the curve's shape. This solver works out all of that from your three coefficients, handling complex roots cleanly when the discriminant is negative so you don't have to do the imaginary arithmetic by hand.

How to use

Enter the three coefficients a, b, and c into the inputs at the top. The equation is rendered back to you in its canonical form so you can sanity-check the signs.

The roots panel shows x₁ and x₂ in the appropriate form — two real numbers, one repeated real number, or a pair of a ± bi — along with a one-line label describing which case you've landed in. Cards beneath show the discriminant, the vertex (x, y), the axis of symmetry, and the y-intercept, plus whether the parabola opens up or down. The copy button beside the roots lifts them as a comma-separated string.

Frequently asked questions

  • What is the quadratic formula?

    For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) / 2a. The solver uses exactly this — the expression under the square root (b² − 4ac) is the discriminant, and its sign decides how many real solutions there are.

  • What does the discriminant tell me?

    It indicates the number and type of roots. Positive discriminant → two distinct real roots, parabola crosses the x-axis at two points. Zero discriminant → one repeated real root, parabola just touches the x-axis. Negative discriminant → no real roots; the parabola never crosses the x-axis, and the two complex roots come as a conjugate pair.

  • What is the vertex and why is it useful?

    The vertex is the highest or lowest point on the parabola — the extreme value of the quadratic. Its x-coordinate is at −b/2a, which is also the axis of symmetry. For physics problems like projectile motion, the vertex gives the peak height; for optimisation, it gives the value of x that maximises or minimises the quadratic.

  • What happens if a is zero?

    The equation isn't quadratic any more — it's linear (bx + c = 0), and has a single solution x = −c/b. The solver flags this as an error rather than producing meaningless output; if you genuinely have a linear equation, just solve it by hand.

  • What does "complex roots" mean in practice?

    It means the parabola never touches the x-axis, so there's no real x value that makes the equation zero. The solver still shows the two complex roots as p ± qi, which are mathematically valid solutions and often physically meaningful (oscillating systems, AC circuits) — but for problems where x must be a real number, "complex roots" means "no solution".

More Maths tools