Solve ax² + bx + c = 0
The quadratic formula gives the roots of ax² + bx + c = 0: x = (-b ± √(b² - 4ac)) / 2a. The part under the root, D = b² - 4ac, is the discriminant. If D is positive there are two real roots, if D is zero there is one repeated root, and if D is negative the roots are a pair of complex numbers. The vertex of the parabola sits at x = -b / 2a.
Example: x² - 3x - 4 = 0 has D = 9 + 16 = 25, so x = (3 ± 5) / 2, which gives x = 4 and x = -1.
Then the equation is linear (bx + c = 0) with the single solution x = -c / b. The solver handles that case too.
Substitute a root back into the equation. For x = 4 in x² - 3x - 4: 16 - 12 - 4 = 0. Products and sums also work: the roots add up to -b/a and multiply to c/a.
When the parabola never crosses the x-axis there is no real solution; the roots involve i, the square root of -1.
See also: square root calculator · fraction calculator · GCF and LCM calculator
Geometry too? Try the Pythagorean theorem calculator.