The Quadratic Formula: How to Use It, With Examples
Quick answer: For any equation ax² + bx + c = 0, the solutions are x = (−b ± √(b² − 4ac)) ÷ 2a. Find a, b and c, work out the discriminant b² − 4ac first, then substitute. If the discriminant is positive there are two solutions, if it's zero there's one, and if it's negative there are no real solutions.
Step by step
Rearrange so one side is 0: ax² + bx + c = 0.
Write down a, b and c, including their signs.
Work out the discriminant, b² − 4ac.
Substitute into the formula and simplify.
What the discriminant tells you
Positive: two real solutions
Zero: one real solution (a repeated root)
Negative: no real solutions
Worked examples
1. Two whole-number answers
Solve x² − 5x + 6 = 0.
a = 1, b = −5, c = 6. Discriminant: 25 − 24 = 1.
x = (5 ± 1) ÷ 2, so x = 3 or x = 2.
2. When a isn't 1
Solve 2x² + 3x − 2 = 0.
Discriminant: 9 − 4(2)(−2) = 9 + 16 = 25.
x = (−3 ± 5) ÷ 4, so x = 0.5 or x = −2.
3. An irrational answer
Solve x² + 4x + 1 = 0.
Discriminant: 16 − 4 = 12.
x = (−4 ± √12) ÷ 2 = −2 ± √3, about −0.27 and −3.73.
4. No real solutions
Solve x² + 2x + 5 = 0.
Discriminant: 4 − 20 = −16. It's negative, so there are no real solutions. The graph never touches the x-axis.
5. One repeated solution
Solve x² − 6x + 9 = 0.
Discriminant: 36 − 36 = 0, so x = 3 only.
A word problem
A ball is thrown so its height in feet after t seconds is h = −16t² + 32t + 48. When does it hit the ground?
Set h = 0: −16t² + 32t + 48 = 0. Divide by −16: t² − 2t − 3 = 0.
Discriminant: 4 + 12 = 16, so t = (2 ± 4) ÷ 2, giving t = 3 or t = −1.
Time can't be negative, so the ball lands after 3 seconds.
When to use something faster
If the equation factors easily, like x² − 5x + 6 = (x − 3)(x − 2), factoring is quicker. The formula always works, so use it when factoring isn't obvious.
Common mistakes
Dropping a negative sign. If b = −5, then −b = 5.
Dividing only the square root by 2a. The whole top line goes over 2a.
Forgetting to rearrange first. x² = 5x − 6 must become x² − 5x + 6 = 0.
When extra support might help
Support is worth considering when algebra steps keep slipping, or before the SAT or PSAT.
A tutor can't guarantee a score, but can make the method automatic. Math Make Smart offers SAT preparation, and you can book a free trial lesson. For straight lines, see our slope-intercept form guide, and for test strategy, our digital SAT math guide.
Frequently asked questions
What is the quadratic formula?
x = (−b ± √(b² − 4ac)) ÷ 2a.
What does the discriminant tell you?
How many real solutions there are: two, one or none.
Can you always use the quadratic formula?
Yes, for any equation in the form ax² + bx + c = 0 with a ≠ 0.