The Quadratic Formula Explained (with a Worked Example)
When a quadratic equation will not factor nicely, the quadratic formula always works. It gives you the solutions to any equation of the form ax² + bx + c = 0.
The formula
For ax² + bx + c = 0, the solutions are: x = (−b ± √(b² − 4ac)) / 2a.
The "±" means you get two answers: one using plus, one using minus.
Worked example
Solve x² + 5x + 6 = 0. Here a = 1, b = 5, c = 6.
x = (−5 ± √(25 − 24)) / 2 = (−5 ± √1) / 2 = (−5 ± 1) / 2.
So x = −2 or x = −3.
The discriminant tells you what to expect
The part under the square root, b² − 4ac, is the "discriminant". Check it first.
- •Positive: two real solutions.
- •Zero: one repeated solution.
- •Negative: no real solutions (the answers are complex).
Frequently asked questions
When should I use the quadratic formula instead of factoring?
Use factoring when the equation factors easily. Use the quadratic formula when it does not — it always works, including for decimals and irrational roots.
What does it mean if the discriminant is negative?
There are no real solutions; the equation has two complex roots. The parabola never crosses the x-axis.