The quadratic formula finds the roots of any quadratic equation ax² + bx + c = 0, even when it won’t factor. The solutions are x = (−b ± √(b² − 4ac)) / 2a, and the part under the root, the discriminant, tells you how many solutions exist.
An algebraic tool that finds the roots (solutions) of any second-degree equation. Start from the standard form, then apply the formula.
the unknown: the solution(s) you're solving for
coefficients from the standard-form equation
gives the two roots: one with +, one with -
the discriminant: sets the number of solutions
Set it equal to zero: ax² + bx + c = 0. Move every term to one side first.
Write down each value, keeping negative signs. a is the x² coefficient, b the x coefficient, c the constant.
Evaluate b² - 4ac: positive → two real solutions; zero → one real solution; negative → two complex solutions.
Plug a, b, c into the formula and simplify. Remember the ± produces two answers: divide the whole numerator by 2a.
The solutions are the x-intercepts of the parabola y = ax² + bx + c, where it crosses the x-axis. The discriminant predicts how many there are.
| DISCRIMINANT | SOLUTIONS | PARABOLA |
|---|---|---|
| b² − 4ac > 0 | Two real solutions | Crosses the x-axis at two points |
| b² − 4ac = 0 | One real solution | Touches the x-axis at the vertex |
| b² − 4ac < 0 | Two complex solutions | Never crosses the x-axis |
Always evaluate the discriminant completely before taking the square root: compute b² - 4ac first.
Watch substituting negatives. -b with b negative becomes positive, and (negative)² is always positive.
Both -b and the ±√ term must be divided by 2a, not just the root term.
It finds the x-intercepts or roots of a quadratic equation. This is especially helpful when the equation cannot be easily factored. It works for every quadratic in the form ax² + bx + c = 0, making it the universal fallback when factoring or completing the square is awkward.
Check the discriminant, b² − 4ac. If it's greater than or equal to zero, the equation has real solutions (two if positive, one if zero). If it's negative, the solutions are complex and the parabola never crosses the x-axis.
Yes. It's highly useful on both the calculator and non-calculator sections of the SAT and ACT. The formula isn't given on the reference sheet, so memorize it. It's the reliable method when an SAT quadratic doesn't factor cleanly, and the discriminant answers how many solutions questions directly.
Factor first when the roots are obviously clean integers, it's faster. Switch to the quadratic formula when the equation doesn't factor with simple whole numbers, when the coefficients are large or messy, or whenever you're unsure. The formula always works; factoring only works sometimes.
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