What is the quadratic formula?
A quadratic equation has an x² term: ax² + bx + c = 0, with a not 0. The quadratic formula solves every one of them. The discriminant, b² − 4ac, tells you what kind of answer to expect before you finish: positive gives 2 real solutions, 0 gives 1, and negative gives 2 complex solutions, written a ± bi.
On a graph, y = ax² + bx + c is a parabola. The real solutions are where it crosses the x-axis. Its turning point, the vertex, sits at x = −b ÷ 2a, and it opens upward when a is positive.
Source OpenStax · Intermediate Algebra 2e, 9.3 Solve Quadratic Equations Using the Quadratic Formula ↗Common Core State Standards · High School Algebra, solve quadratic equations (HSA-REI.B.4b) ↗OpenStax · Intermediate Algebra 2e, 9.6 Graph Quadratic Functions Using Properties ↗
How to use the quadratic formula
Here it is with your numbers:
- Discriminant: 9² − 4 × 2 × (−5) = 121
- x = (−9 ± √121) ÷ 4
- x = (−9 ± 11) ÷ 4
- x = −5 or 0.5
It means the parabola crosses the x-axis at −5 and 0.5.
Quadratic formula examples
Worked examples from OpenStax. Tap one to try it.
| Equation | Solutions |
|---|---|
| 2x² + 9x − 5 = 0 | −5 or 1/2 |
| x² − 6x + 5 = 0 | 1 or 5 |
| 2x² + 10x + 11 = 0 | (−5 ± √3) ÷ 2 |
| 4x² − 20x + 25 = 0 | 5/2 |
| 3x² + 2x + 9 = 0 | (−1 ± i√26) ÷ 3 |
| x² − 49 = 0 | −7 or 7 |
FAQs
What is the formula for solving quadratics?
For ax² + bx + c = 0, where a isn’t 0, the quadratic formula is x = (−b ± √(b² − 4ac)) ÷ 2a. The ± gives the two solutions: one with plus, one with minus.
Source OpenStax · Intermediate Algebra 2e, 9.3 Solve Quadratic Equations Using the Quadratic Formula ↗
How to find the value of discriminant?
The discriminant is b² − 4ac, the part under the square root. For 2x² + 9x − 5 = 0 it is 9² − 4 × 2 × (−5), which is 81 + 40, or 121. Above 0 means 2 real solutions, 0 means 1, and below 0 means 2 complex solutions.
What if discriminant is 0?
Then the equation has exactly 1 real solution, x = −b ÷ 2a, because adding or subtracting √0 changes nothing. For 4x² − 20x + 25 = 0 that is 20 ÷ 8, which is 5/2.
What is the easiest method in solving a quadratic equation?
OpenStax’s advice: try factoring first, because it is often the quickest. Use the square root property when the equation is ax² = k, and the quadratic formula for anything else. The formula works on every quadratic equation, but it isn’t always the easiest.
Sources
- OpenStax: Intermediate Algebra 2e, 9.3 Solve Quadratic Equations Using the Quadratic Formula
- OpenStax: Intermediate Algebra 2e, 9.6 Graph Quadratic Functions Using Properties
- Common Core State Standards: High School Algebra, solve quadratic equations (HSA-REI.B.4b)