How the quadratic formula works
A quadratic equation is any equation that can be written in the standard form ax² + bx + c = 0, where a, b and c are numbers and a is not zero. Its graph is a parabola, and the roots — the values of x that make the equation true — are exactly where that parabola crosses the x-axis.
Every quadratic is solved by the same tool, the quadratic formula: x = (−b ± √(b²−4ac)) / 2a. The expression under the square root, b²−4ac, is called the discriminant. Compute it first: its sign tells you whether you will get two real roots, one repeated root, or a pair of complex conjugates, before you finish the arithmetic.
The sign of the discriminant fully determines the number and type of solutions:
| Discriminant b²−4ac | Number of roots | Type of roots |
|---|---|---|
| Positive (> 0) | 2 | Two distinct real roots |
| Zero (= 0) | 1 | One repeated real root (a double root) |
| Negative (< 0) | 2 | Two complex conjugate roots, p ± qi |
Reference note: the vertex of y = ax² + bx + c sits at (−b/2a, c − b²/4a) and the vertical line x = −b/2a is the axis of symmetry. If a = 0 the equation is linear, bx + c = 0, and this tool solves it as x = −c/b instead.
Frequently asked questions
- What is the quadratic formula?
- It is x = (−b ± √(b²−4ac)) / 2a, which gives both solutions of any equation written as ax² + bx + c = 0 with a not zero. The ± means you compute one root with a plus and the other with a minus.
- How do I solve a quadratic equation?
- Put it in standard form ax² + bx + c = 0, read off a, b and c, then substitute into the formula. Work out the discriminant b²−4ac first, take its square root, and evaluate the plus and minus cases to get both roots.
- What is the discriminant?
- The discriminant is D = b²−4ac, the value under the square root. Its sign tells you the roots: positive gives two distinct real roots, zero gives one repeated real root, and negative gives two complex conjugate roots.
- What does it mean when the discriminant is negative?
- A negative discriminant has an imaginary square root, so there are no real solutions. Instead you get two complex conjugate roots of the form p ± qi, with p = −b/2a and q = √(4ac−b²)/2a.
- How do I find the vertex of a parabola?
- For y = ax² + bx + c the vertex is at x = −b/2a, and its y-value is c − b²/4a. The line x = −b/2a is the axis of symmetry; the vertex is a minimum when a is positive and a maximum when a is negative.
- What if a = 0?
- Then the equation is linear, bx + c = 0, not quadratic. If b is not zero it has the single solution x = −c/b; if both a and b are zero there is either no solution or every value works, depending on c.