Quadratic Formula: When Factoring Fails

The quadratic formula solves any equation of the form ax^2 + bx + c = 0. The expression b^2 - 4ac under the square root is called the discriminant. If discriminant > 0: two distinct real roots (parabola crosses x-axis twice). If = 0: one repeated root (parabola touches axis). If < 0: two complex conjugate roots (parabola never touches axis). The formula works even when factoring is impossible — try x^2 + 3x + 1 = 0 with discriminant 9-4=5, giving x = (-3 +/- sqrt(5)) / 2. Use our Quadratic Solver to check your work.