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.
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