How do you use the quadratic formula?

For ax^2 + bx + c = 0, plug a, b, and c into x = (-b +/- sqrt(b^2 - 4ac)) / 2a. For x^2 - 5x + 6 = 0: a = 1, b = -5, c = 6, so x = (5 +/- sqrt(25 - 24)) / 2 = (5 +/- 1) / 2, giving x = 3 or x = 2.

What is the discriminant?

The discriminant is b^2 - 4ac, the part under the square root in the quadratic formula. It tells you the nature of the roots before you finish solving.

What if the discriminant is negative?

A negative discriminant means no real roots: the parabola never crosses the x-axis. The solutions are a complex conjugate pair. For x^2 + 1 = 0, the discriminant is -4 and the roots are x = i and x = -i.

Quadratic Formula Calculator

Q: How do you solve a quadratic with complex roots?

Use the formula as normal — the square root of the negative discriminant becomes an imaginary part. For x²+1=0: x = ±i. The calculator displays these in a ± bi form.

Q: What if a = 0?

Then it is not quadratic at all — it is the linear equation bx + c = 0, with the single solution x = −c/b. The calculator detects this and solves the linear equation instead of erroring.

Solve any quadratic equation ax² + bx + c = 0. Enter the three coefficients, the calculator finds the roots (real or complex) and shows every substitution.

How to use this calculator

Enter the coefficients a, b, and c of your equation and press Calculate. The calculator evaluates the discriminant, applies the quadratic formula, and shows each substitution, including complex roots when the discriminant is negative. If you enter a = 0, it honestly drops back to the linear equation instead of dividing by zero.

How it works

The quadratic formula solves every quadratic: x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the root, b² − 4ac, is the discriminant: positive means two real roots, zero means one repeated root, negative means two complex roots. The calculator checks the discriminant first, so it always takes the right path.

x = (−b ± √(b² − 4ac)) ÷ 2a

Worked example

Example: x² − 5x + 6 = 0

Step 1, Discriminant: D = (−5)² − 4(1)(6) = 25 − 24 = 1. Positive → two real roots.

Step 2, Formula: x = (5 ± √1) ÷ 2.

Step 3, Roots: x₁ = (5+1)/2 = 3, x₂ = (5−1)/2 = 2.

Check: (3)² − 5(3) + 6 = 0 ✓ and (2)² − 5(2) + 6 = 0 ✓.

Common mistakes

Sign errors on −b: if b = −5, then −b = +5. Most wrong answers trace back to this one flip.
Forgetting ÷ 2a applies to everything: the denominator covers the whole (−b ± √D), not just the root.
Assuming “no real roots” means “no roots”: a negative discriminant still gives two complex roots, the calculator shows them in a ± bi form.

When to use it

Quadratics model anything with acceleration or area: projectile motion, revenue-vs-price curves, area optimization, and the characteristic equations behind spring and circuit behavior. Whenever the highest power of x is 2, this is the tool.

For a complete worked example with a negative discriminant, see complex roots solved step by step for x²+4x+5=0.

Frequently asked questions

What is the quadratic formula?

x = (−b ± √(b²−4ac)) ÷ 2a. It gives the solutions of any equation ax² + bx + c = 0 directly from its three coefficients.

What does the discriminant tell you?

The sign of b²−4ac predicts the roots: positive → two distinct real roots; zero → one repeated real root; negative → two complex conjugate roots.

How do you solve a quadratic with complex roots?

Use the formula as normal, the square root of the negative discriminant becomes an imaginary part. For x²+1=0: x = ±i. The calculator displays these in a ± bi form.

What if a = 0?

Then it is not quadratic at all, it is the linear equation bx + c = 0, with the single solution x = −c/b. The calculator detects this and solves the linear equation instead of erroring.