How to solve a quadratic equation with the quadratic formula
Write the equation in standard form ax² + bx + c = 0, then read off a, b and c. Signs matter: in x² − 5x + 6 = 0, b is −5, not 5.
Substitute into x = (−b ± √(b² − 4ac)) / 2a. The ± gives two answers: one with + and one with −.
Simplify the discriminant b² − 4ac first. It tells you what kind of answer to expect before you finish.
What the discriminant b² − 4ac tells you
Positive: two different real roots. Example: x² − 5x + 6 = 0 has discriminant 1, so x = 3 or x = 2.
Zero: one repeated real root. Example: x² − 6x + 9 = 0 has discriminant 0, so x = 3 (the parabola just touches the x-axis).
Negative: no real roots, two complex roots. Example: x² + 2x + 5 = 0 has discriminant −16, so x = (−2 ± 4i) / 2 = −1 ± 2i.
Quadratic formula vs. factoring vs. completing the square
Factoring is fastest when the roots are whole numbers: x² − 5x + 6 = (x − 3)(x − 2).
Completing the square is how the quadratic formula is derived, and is useful for vertex form y = a(x − h)² + k.
The quadratic formula always works, including irrational and complex roots, which is why this solver uses it.
