Solving Quadratic Inequalities
Simple Explanation
To solve a quadratic inequality like ax² + bx + c > 0, first find the roots of the related equation ax² + bx + c = 0 (using factoring or the quadratic formula), then use those roots to divide the number line into intervals and test a point from each interval to see whether it satisfies the inequality.
Why Do We Need It?
Quadratic inequalities answer "for which values of x is this quantity positive/negative?" — a question that comes up whenever you need a range of valid values, not just a single answer.
See It
An upward-opening parabola crossing the x-axis at x = -2 and x = 3, showing the graph is below the axis between the roots and above the axis outside them
Formula
The Quadratic Formula
x = (−b ± √(b² − 4ac)) / 2a
Solves any quadratic equation ax² + bx + c = 0 directly from its coefficients, without needing to factor.
- x
- — the solution(s) — the roots of the equation
- a, b, c
- — the coefficients of ax² + bx + c = 0 (a ≠ 0)
- ±
- — produces two solutions — one using +, one using −, unless the discriminant is 0
When to use it: Whenever a quadratic equation cannot be factored easily (or at all), or you want a reliable method that always works.
Worked Example
Solve a quadratic inequality
Solve x² − x − 6 > 0.
Why Does This Work?
Since the graph only changes from positive to negative (or back) at its roots, the sign of ax² + bx + c must stay constant across any interval between consecutive roots — testing one point per interval is enough to determine the sign for the entire interval.
Real-Life Example
Finding a profitable price range
A company's profit is modeled by a downward-opening quadratic in terms of price, and the company wants to know which prices keep profit positive.
Solving "profit(price) > 0" as a quadratic inequality gives the exact range of prices that keep the company profitable — not just one break-even value.
Practice
Solve x² − 9 < 0.
HardCommon mistake
Including or excluding the roots incorrectly for a strict versus non-strict inequality — a strict inequality (< or >) excludes the roots themselves, while ≤ or ≥ includes them.
Quick Review
- Find the roots of the related equation first — they split the number line into intervals.
- Test one point from each interval to check its sign.
- Collect every interval whose sign matches the inequality.