Quadratic Formula Solver

Solve ax^2 + bx + c = 0, show the discriminant, return real or complex roots, and locate the parabola's vertex.

Last updated: 2026-03-25

Quadratic formula solver

Enter your values

Enter the a, b, and c coefficients for ax^2 + bx + c = 0 and solve the equation step by step.

Roots

Enter the coefficients of a quadratic equation to compute its roots, discriminant, and vertex.

Calculation History(0)

No calculations yet. Complete a calculation to see it here.

Example calculations

Tap an example to prefill the calculator with sample values.

Two real roots

x^2 - 5x + 6 = 0

A classic factoring example that also shows the discriminant staying positive.

Result: The parabola opens upward and crosses the x-axis twice.

Complex roots

2x^2 + 4x + 5 = 0

Useful when the discriminant is negative and the roots become complex.

Result: The roots have the same real part with opposite imaginary parts.

How the quadratic solver works

The solver computes the discriminant b^2 - 4ac first. That value determines whether the roots are distinct real numbers, a repeated real root, or a complex pair.

It also computes the axis of symmetry and vertex so you can understand the parabola's shape, not just the raw x-intercepts.

Quadratic solver FAQs

A short guide to discriminants, repeated roots, and why the vertex still matters.

Why does the discriminant matter?

The discriminant tells you how many real solutions the quadratic has. Positive means two real roots, zero means one repeated root, and negative means the roots are complex.

What if a = 0?

Then the equation is not quadratic anymore. This solver expects a non-zero a coefficient so the parabola and quadratic formula remain valid.

Embed this calculator

Copy the code below to embed this calculator on your website or blog. It's free — no API key needed.

Iframe embed code

<iframe
  src="https://calc.mintloop.dev/embed/education/quadratic-formula-solver"
  width="100%"
  height="600"
  frameborder="0"
  title="Quadratic Formula Solver"
  loading="lazy">
</iframe>

Optional: auto-resize script

<script>
var CALC_HUB_ORIGIN = 'https://calc.mintloop.dev';
window.addEventListener('message', function(e) {
  if (e.origin !== CALC_HUB_ORIGIN) return;
  if (!e.data || e.data.type !== 'calc-hub-resize') return;
  var frames = document.querySelectorAll('iframe[src*="calc.mintloop.dev"]');
  frames.forEach(function(f) {
    if (f.contentWindow === e.source) {
      f.style.height = String(Math.max(0, Number(e.data.height) || 0)) + 'px';
    }
  });
});
</script>

Stay in study mode with nearby trig and academic calculators.

Trig Unit Circle Calculator

Switch from algebra practice to exact trig values, quadrants, and radian work.

GPA Calculator

Use another academic planning tool when you move from homework to grade tracking.

Get more education calculators

Join the Calc Hub newsletter for new calculators covering algebra, trig, unit conversions, and classroom study tools.

Join the Calc Hub newsletter

Was this calculator helpful?

Your feedback helps us improve future calculators.