The quadratic formula
A quadratic equation has the form ax² + bx + c = 0, with a ≠ 0. Its solutions are given by x = (−b ± √(b² − 4ac)) ÷ 2a.
The discriminant
The expression b² − 4ac is called the discriminant and tells you the type of solutions:
- Positive: two different real roots.
- Zero: one repeated real root.
- Negative: two complex conjugate roots.
Example
For x² − 3x + 2 = 0: the discriminant is 9 − 8 = 1, so x = (3 ± 1) ÷ 2, giving x = 2 and x = 1.
Frequently asked questions
What if a = 0?
Then the equation is linear, not quadratic, and has one solution x = −c/b.
What are complex roots?
Roots that involve the imaginary unit i, where i² = −1. They appear when the parabola does not cross the x-axis.
Can I use decimals?
Yes, coefficients can be any real numbers.