Every quadratic equation in the standard form ax² + bx + c = 0 can be solved using the same formula, regardless of the specific coefficients — and a single value calculated partway through, called the discriminant, tells you in advance what kind of solution to expect before you even finish solving.
The quadratic formula works because it's derived from completing the square on the general equation — a proof that holds for any values of a, b, and c (as long as a isn't zero, which would make the equation linear, not quadratic). The discriminant, b² − 4ac, determines whether the equation has two real solutions, one repeated real solution, or two complex solutions.
The Quadratic Formula
x = [−b ± √(b² − 4ac)] ÷ 2a
Discriminant = b² − 4ac
If the discriminant is positive, there are two distinct real roots. If it's exactly zero, there's one repeated real root. If it's negative, the roots are complex (involving i, the imaginary unit), since you can't take the square root of a negative number within the real number system.
A Worked Example
For x² − 3x + 2 = 0 (a=1, b=−3, c=2): Discriminant = (−3)² − 4(1)(2) = 9 − 8 = 1, positive, so two real roots. x = [3 ± √1] ÷ 2 = (3 ± 1) ÷ 2, giving x = 2 or x = 1.
For x² + 2x + 5 = 0 (a=1, b=2, c=5): Discriminant = 4 − 20 = −16, negative, so complex roots. x = [−2 ± √−16] ÷ 2 = −1 ± 2i.
Common Mistakes to Avoid
- Forgetting the ± symbol produces two separate answers: most quadratic equations have two solutions, and both need to be calculated and reported.
- Making sign errors with negative coefficients: careful attention to negative values of b and c is one of the most common sources of quadratic formula errors.
- Not checking the discriminant before assuming real solutions exist: a negative discriminant means the solutions are complex, not an error in your calculation.
- Confusing the quadratic formula with factoring shortcuts: factoring works for some quadratics but not all — the quadratic formula always works, regardless of whether the equation factors neatly.
Bottom Line
The quadratic formula solves any equation in ax² + bx + c = 0 form, real or complex roots included. Use a Quadratic Equation Solver to instantly solve any quadratic and see exactly which type of roots it produces.