Quadratic Equation Calculator

Q: What is the discriminant and why does it matter?

It is b² − 4ac. Positive gives two real roots, zero gives one repeated root, and negative means the roots are complex with no real solution.

Q: What if my equation has no real roots?

The parabola never crosses the x-axis. The solutions are complex numbers involving the square root of a negative, which this tool reports as 'no real roots.'

Q: Can I always factor instead of using the formula?

Only when the roots are nice numbers. The quadratic formula works for every quadratic, including those that do not factor neatly.

Real roots of ax^2 + bx + c = 0.

Input sheet

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Quadratic equations describe projectile paths, profit-maximizing prices, and many geometry problems. This solves ax² + bx + c = 0 for its real roots using the quadratic formula.

How it works

Quadratic formula: x = (−b ± sqrt(b² − 4ac)) ÷ 2a. The discriminant decides if roots are real.

The discriminant, b² − 4ac, tells you what to expect before solving: positive means two distinct real roots, zero means one repeated root, and negative means the roots are complex (this tool reports no real roots in that case).

The two roots are where the parabola crosses the x-axis. The leading coefficient a sets whether the parabola opens upward (a > 0) or downward (a < 0); if a is zero the equation is not quadratic at all.

x = (−b ± square root of (b² − 4ac)) ÷ (2a)

Worked examples

Solving x² − 5x + 6 = 0 (a=1, b=−5, c=6). → x = 3 and x = 2
Discriminant is 25 − 24 = 1; roots are (5 ± 1) ÷ 2, giving 3 and 2.
Solving x² − 4x + 4 = 0 (a=1, b=−4, c=4). → x = 2 (repeated)
Discriminant is 16 − 16 = 0, so both roots equal (4 ± 0) ÷ 2 = 2.

Tips & gotchas

Check the discriminant first: if b² − 4ac is negative, there are no real solutions.
The two roots sum to −b/a and multiply to c/a — a quick way to verify your answers.
If a equals zero, the equation is linear, not quadratic; solve bx + c = 0 directly.

FAQ

What is the discriminant and why does it matter?

It is b² − 4ac. Positive gives two real roots, zero gives one repeated root, and negative means the roots are complex with no real solution.

What if my equation has no real roots?

The parabola never crosses the x-axis. The solutions are complex numbers involving the square root of a negative, which this tool reports as 'no real roots.'

Can I always factor instead of using the formula?

Only when the roots are nice numbers. The quadratic formula works for every quadratic, including those that do not factor neatly.

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