Quadratic Equation Solver
Solve ax² + bx + c = 0 — real or complex roots, discriminant and vertex.
How to use the Quadratic Equation Solver
- Enter the coefficients a, b and c.
- Read the roots — real or complex.
- Check the discriminant and vertex below.
x = (−b ± √(b²−4ac)) ÷ 2a
About the Quadratic Equation Solver
This solver takes the three coefficients of a quadratic, ax² + bx + c = 0, and applies the quadratic formula x = (−b ± √(b²−4ac)) ÷ 2a. The quantity under the square root, b²−4ac, is the discriminant, and it decides the type of answer: positive gives two distinct real roots, zero gives one repeated real root, and negative gives a complex conjugate pair written a ± bi. Since a quadratic needs an x² term, the tool rejects a = 0 as a linear equation rather than a quadratic.
It also reports the vertex at (−b/2a, f(−b/2a)) — the parabola’s turning point, its lowest point when a is positive and its highest when a is negative. For example, x² − 5x + 6 = 0 has discriminant 25 − 24 = 1, roots x = 2 and x = 3, and a vertex at (2.5, −0.25).
It suits homework, physics problems and anywhere a parabola appears. One subtlety: a negative discriminant simply means the curve never crosses the x-axis, so there is no real solution. To take roots on their own, use the root calculator, or raise numbers to powers with the exponent calculator.
Frequently asked questions
What does the discriminant tell me?
The discriminant is b²−4ac, the value under the square root. Positive means two different real roots, zero means one repeated real root, and negative means two complex roots — so you can tell the type of answer before you finish solving.
What is the vertex?
The vertex is the turning point of the parabola, at x = −b/2a with its matching y-value. When a is positive the vertex is the lowest point of the curve; when a is negative it is the highest. It also sits on the axis of symmetry.
Why will it not accept a = 0?
With a = 0 there is no x² term, so ax² + bx + c = 0 collapses to the linear equation bx + c = 0, which is solved differently. A quadratic needs a non-zero a, so the tool asks you to change it.
What do complex roots mean?
When the discriminant is negative the parabola never crosses the x-axis, so there is no real solution. The answers are then a pair of complex numbers of the form −b/2a ± (√(4ac−b²) ÷ 2a)i, shown as a ± bi.
Can it handle decimals and negative coefficients?
Yes. Enter any real numbers for a, b and c, including decimals and negatives. The formula works the same way, and results are rounded to four decimal places so they stay readable.