Quadratic Equation Solver

Quadratic equations are a core part of Class 10 maths and every competitive exam. This quadratic equation solver finds the roots of ax² + bx + c = 0 using the quadratic formula, tells you their nature from the discriminant, gives the vertex of the parabola, and draws the graph so you can see where it crosses the x-axis. Complex roots are shown in p ± qi form.

How to use it

Enter the coefficients a, b and c (use negative numbers for minus signs).
Read the roots, the discriminant and the working.

Example

For x² − 5x + 6 = 0: a = 1, b = −5, c = 6, so D = 25 − 24 = 1 > 0 — two real roots. x = (5 ± 1) ÷ 2, giving x = 3 and x = 2, which matches the factorisation (x − 2)(x − 3). For x² + 2x + 5 = 0, D = −16, so the roots are complex: −1 ± 2i. Plot other functions with the graph plotter, or do the arithmetic in the scientific calculator.

Frequently asked questions

What is the quadratic formula?

For ax² + bx + c = 0, the roots are x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, D = b² − 4ac, is called the discriminant.

What does the discriminant tell me?

If D > 0 there are two different real roots; if D = 0, two equal (repeated) real roots; if D < 0, no real roots — the roots are complex numbers p ± qi.

What is the vertex of a parabola?

The turning point of the graph y = ax² + bx + c, at x = −b ÷ 2a. It is the minimum point when a > 0 and the maximum when a < 0.

What if a = 0?

Then the equation is linear, bx + c = 0, with the single root x = −c ÷ b. The solver handles this case too.