Quadratic Inequality

1Summary 重點摘要

Solve quadratic inequalities, use opening up/down method.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Solve the related equationTo solve a quadratic inequality, first pretend the inequality sign is an equals sign and solve the related quadratic equation to find its two roots. These roots are the critical values on the number line, because the sign of a quadratic expression can only change where it passes through a root. Factorisation or the quadratic formula both work, exactly as in solving an equation. A common mistake is finding roots before converting to standard form, which leads to wrong coefficients.
2. Step 2: Decide the direction of openingNext look at the sign of the quadratic coefficient a to decide how the parabola opens: when a is positive it opens upward like a bowl, and when a is negative it opens downward, the other way up. The direction decides which regions give values above 0 and which give values below 0, so it must be settled first. In standard form, a is simply the coefficient of the x² term. A common mistake is misreading the sign, for example treating -x² as x².
3. Step 3: Mark the roots on a number lineMark the two roots on a number line, which is now split into three regions: left of both roots, between them, and right of both. Since the sign of a quadratic only changes at a root, the sign is constant within each region, so testing one point per region is enough. When drawing endpoints, watch the strictness of the inequality: use open circles for > or <, and solid dots for ≥ or ≤. A common mistake is swapping the positions of the two roots.
4. Step 4: Judge when the curve opens upwardWhen the parabola opens upward, its two ends rise without bound while the middle dips below the x-axis. Hence the solution of the quadratic being greater than 0 lies in the two outer regions, and the solution of it being less than 0 lies between the roots. For example, x² + 5x + 6 > 0 gives x smaller than the smaller root or larger than the larger root. Test points such as 0 are quickest. A common mistake is mixing up the outer and middle regions, which inverts the answer.
5. Step 5: Judge when the curve opens downwardWhen the parabola opens downward the situation reverses: the middle rises above the x-axis while the two ends fall away. Hence the expression is greater than 0 between the roots and less than 0 outside them. Do not memorise blindly; sketch a quick curve and read the answer from where the graph sits relative to the x-axis, so both cases stay clear. A common mistake is copying the upward-opening conclusion without adjusting for the direction.
6. Step 6: Write the solution setFinally write the answer as intervals and decide whether endpoints are included: for strict > or < the endpoints are excluded, written as x < a or x > b; for ≥ or ≤ the endpoints are included and the symbols must show it. A two-sided answer needs the word or to join the parts, while a middle region is written as one interval. A common mistake is dropping endpoints in an inclusive problem, or writing two outer regions as a single interval.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• Solve inequality first solve equation find roots
• See quadratic term positive/negative judge opening
• Opens up: > 0 two sides, < 0 middle
• Opens down: > 0 middle, < 0 two sides
• Multiply/divide negative must flip sign