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Quadratic Formula Solver

Solve any quadratic equation ax² + bx + c = 0 step by step.

What is the Quadratic Formula Solver?

A quadratic formula solver finds the values of x that satisfy an equation in the form ax² + bx + c = 0, using the standard quadratic formula x = (−b ± √(b² − 4ac)) ÷ 2a. It works by first calculating the discriminant (b² − 4ac), which determines the nature of the roots: a positive discriminant gives two distinct real roots, a discriminant of exactly zero gives one repeated real root, and a negative discriminant gives two complex (imaginary) roots. To solve one by hand, calculate the discriminant first, then substitute it into the formula above along with a, b and c — if the discriminant is negative, take the square root of its absolute value and express the result as a ± bi using the imaginary unit i.

How it works

Any quadratic equation in the form ax² + bx + c = 0 can be solved using the quadratic formula: x = (−b ± √(b² − 4ac)) / 2a. The ± symbol means there are generally two solutions (roots) — one using addition, one using subtraction — which correspond to the two points where the equation's graph (a parabola) crosses the x-axis.

UK context

The expression under the square root, b² − 4ac, is called the discriminant, and it tells you what kind of solutions to expect before you even finish solving: a positive discriminant means two distinct real roots, a discriminant of exactly zero means one repeated real root (the parabola just touches the x-axis at a single point), and a negative discriminant means no real roots — the solutions exist, but only as complex numbers.

Tips

Double-check the sign of b before substituting into the formula — a sign error here is the single most common mistake when solving quadratics by hand.
Calculate the discriminant (b² − 4ac) first as a sanity check — a negative value tells you immediately that there are no real-number solutions, before you go further.
If a = 0, the equation isn't actually quadratic (there's no x² term) — it becomes a simple linear equation instead, and the quadratic formula doesn't apply.

Frequently asked questions

What is the quadratic formula?

For an equation in the form ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) / 2a, which gives the value(s) of x that satisfy the equation.

What does the discriminant tell you?

The discriminant is the part of the formula under the square root, b² − 4ac. If it's positive, there are two distinct real roots. If it's zero, there's exactly one repeated real root. If it's negative, there are no real roots (only complex ones).

Can a quadratic equation have no real solutions?

Yes — when the discriminant (b² − 4ac) is negative, the equation has no real-number solutions, because you'd need to take the square root of a negative number. The graph of the equation simply doesn't cross the x-axis.