📐 Quadratic Formula Solver
Solve ax² + bx + c = 0 with steps
The Quadratic Formula
For any equation in the form ax² + bx + c = 0, the solutions are: x = (-b ± √(b²-4ac)) / 2a. The expression under the square root, b²-4ac, is called the discriminant. If the discriminant is positive, there are two real solutions. If zero, one real solution (a repeated root). If negative, two complex (imaginary) solutions.
The vertex (highest or lowest point) of the parabola is at x = -b/2a, y = f(-b/2a). If a is positive, the parabola opens upward (vertex is a minimum). If a is negative, it opens downward (vertex is a maximum). The axis of symmetry passes through the vertex at x = -b/2a.
When to use the quadratic formula vs factoring
If the equation factors easily (x²-5x+6 = (x-2)(x-3)), factoring is faster. Use the quadratic formula when factoring is not obvious, when coefficients are large or non-integer, or when the discriminant is negative (complex roots cannot be found by factoring). The quadratic formula always works — factoring only works when roots are rational.