Skip to main content
Book Consultation →
Algebra100% FreeInstant

Quadratic Equation Solver: Roots, Discriminant and Vertex

This free quadratic equation solver finds the roots of ax² + bx + c = 0. It shows the discriminant, tells you whether the roots are real or complex, and gives the vertex and axis of symmetry.

By Debmedia Technologies LLP · Kolkata, India · Updated

What Is It and When Do You Need It?

A quadratic equation has the form ax² + bx + c = 0 with a not equal to zero. Quadratics describe the path of a thrown ball, the area of a rectangle with changing sides, profit curves and many school problems. The answers, called roots, are the x values where the curve crosses the x axis.

This solver gives the roots even when they are complex numbers, and adds the vertex and the axis of symmetry so you can sketch the parabola.

How to Use This Tool

  1. Type the coefficients a, b and c. Decimals and negatives are fine.
  2. Press Calculate.
  3. Read the roots and the discriminant.
  4. Use the vertex and axis to sketch or check the graph.

How the Calculation Works

The discriminant is D = b² − 4ac. If D ≥ 0 the roots are (−b ± √D) ÷ 2a. If D < 0 the roots are complex: −b ÷ 2a ± i√(−D) ÷ 2a. The sum of the roots is always −b ÷ a and their product is c ÷ a, which the tool prints as a quick check.

The vertex is at x = −b ÷ 2a and y = c − b² ÷ 4a.

Worked Example

Solve x² − 3x + 2 = 0. Here a = 1, b = −3 and c = 2, so D = 9 − 8 = 1. The roots are (3 ± 1) ÷ 2, which gives x = 2 and x = 1. The vertex is at x = 1.5 and y = −0.25.

Solve x² + 2x + 5 = 0. D = 4 − 20 = −16, so the roots are complex: −1 ± 2i.

Common Uses

  • Algebra homework and exam revision.
  • Projectile motion in physics.
  • Finding break-even points from a profit formula.
  • Checking hand-worked answers.

Tips and Common Mistakes

  • Move every term to one side first so the right side is zero.
  • Check the answer: the roots should add up to −b ÷ a.
  • If D is a perfect square, the equation factors neatly.

Next, you may need the exponent and root calculator, logarithm calculator, distance, midpoint and slope calculator, area calculator.

Everything runs in your browser. Nothing you type is uploaded, stored or sent to any outside service.

Frequently Asked Questions

For ax² + bx + c = 0 the roots are x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, b² − 4ac, is the discriminant.
If it is positive there are two different real roots. If it is zero there is one repeated root. If it is negative the roots are complex numbers.
Then the equation is not quadratic. It becomes the straight-line equation bx + c = 0, and the tool solves that instead.
The vertex is the turning point of the parabola y = ax² + bx + c. Its x value is −b ÷ 2a. It is the lowest point if a is positive and the highest if a is negative.

Disclaimer: results are calculated in your browser from the numbers you enter. Check important results before you rely on them for money, health, safety or legal decisions.