Quadratic Formula Calculator
Solve any quadratic equation ax² + bx + c = 0 using the quadratic formula.
The Formula
For ax² + bx + c = 0
Discriminant: b² − 4ac
> 0: two real roots · = 0: one root · < 0: no real roots
Discriminant = 25 + 24 = 49
x = (−5 ± √49) / 4 = (−5 ± 7) / 4
x = 0.5 or x = −3
The Quadratic Formula
Any equation of the form ax² + bx + c = 0 (where a is not zero) can be solved with the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The ± symbol means the formula produces two solutions — one using addition and one using subtraction. These two solutions are the x-intercepts of the parabola y = ax² + bx + c, also called its roots or zeros.
The Discriminant: Predicting the Nature of Solutions
The expression b² - 4ac under the square root is called the discriminant, and it tells you what kind of solutions to expect before you do the full calculation. If the discriminant is positive, there are two distinct real roots — the parabola crosses the x-axis at two points. If it is zero, there is exactly one real root (a repeated root) — the parabola just touches the x-axis. If it is negative, there are no real solutions, only two complex conjugate roots — the parabola does not intersect the x-axis at all.
When to Use the Quadratic Formula vs. Factoring
Factoring is faster when the quadratic factors neatly — like x² - 5x + 6 = (x-2)(x-3). But many quadratics do not factor over integers. The quadratic formula always works, for any values of a, b, and c, regardless of whether the roots are integers, fractions, irrational numbers, or complex. If you are unsure whether a quadratic factors neatly, the quadratic formula is the reliable fallback.
Frequently Asked Questions
What is the quadratic formula used for?
The quadratic formula is used to find the x-intercepts (roots) of a parabola, solve projectile motion problems, calculate optimal values in business (break-even, max profit), and solve countless engineering and physics problems.
Are there other ways to solve quadratic equations?
Yes: factoring (when factors are integers), completing the square, and graphing. The quadratic formula always works and is the most reliable method.