Quadratic Equation Solver

The quadratic formula

x = (−b ± √(b² − 4ac)) ÷ 2a. The part under the square root, b² − 4ac, is the discriminant:

• Greater than 0: two different real roots (the parabola crosses the x-axis twice).
• Equal to 0: one repeated root (it just touches the axis).
• Less than 0: no real roots, only two complex roots of the form p ± qi.

Example

x² − 3x − 10 = 0 has a = 1, b = −3, c = −10. The discriminant is 9 + 40 = 49, so x = (3 ± 7) ÷ 2, giving x = 5 or x = −2. Check: (5)² − 3(5) − 10 = 0.

The graph

The vertex at x = −b ÷ 2a is the lowest point of the curve when a is positive, and the highest when a is negative. The sum of the roots is −b/a and their product is c/a, a quick way to check answers in exams. If a = 0, the equation is linear and has the single solution x = −c/b.