Quadratic Formula Calculator
Enter the coefficients of any quadratic equation in the form ax² + bx + c = 0 to find its roots, including complex roots when the discriminant is negative.
Show solution steps
What is the Quadratic Formula Calculator?
A quadratic equation is any equation of the form ax² + bx + c = 0, where the highest power of x is 2. Graphed, it always forms a parabola — the U-shaped curve you see in projectile motion (a ball thrown in the air), in profit/area optimisation problems, and throughout physics and engineering wherever a quantity depends on the square of another.
How to Calculate
Read off the coefficients a, b and c directly from your equation in standard form, and enter them as they are (including their sign). The calculator first works out the discriminant, b² − 4ac, which tells you what kind of roots to expect — two distinct real roots if it's positive, one repeated real root if it's exactly zero, or two complex roots if it's negative — before computing the actual root values.
Formula
x = (−b ± √(b2 − 4ac)) / 2a
The discriminant b² − 4ac determines the nature of the roots: positive gives two real roots, zero gives one repeated real root, and negative gives two complex roots.
Worked Example
For x² − 3x + 2 = 0 (a=1, b=−3, c=2): discriminant = 1, roots are x=2 and x=1.
How to Use
- Enter the coefficients a, b, and c.
- Read the discriminant and both roots instantly.
- If the discriminant is negative, the roots are shown in complex number form (real u00b1 imaginary i).