2.2.5 Quadratic Inequalities

Solve quadratic inequalities and represent solutions using interval notation and number line graphs.

定义

A quadratic inequality is an inequality that involves a quadratic expression, typically written in one of the following forms: \(ax^2 + bx + c > 0\), \(ax^2 + bx + c < 0\), \(ax^2 + bx + c \geq 0\), or \(ax^2 + bx + c \leq 0\), where \(a \neq 0\). To solve a quadratic inequality, we find the roots of the corresponding quadratic equation \(ax^2 + bx + c = 0\) using factoring, the quadratic formula, or completing the square. These roots divide the number line into intervals. We then test a point from each interval to determine which intervals satisfy the inequality. The solution set is expressed using interval notation (such as \((a, b)\), \([a, b]\), \((a, b]\), or \([a, b)\)) or represented graphically on a number line. When \(a > 0\), the parabola opens upward; when \(a < 0\), it opens downward, which affects which intervals satisfy the inequality.

核心公式

• \(ax^2 + bx + c > 0 \text{ or } ax^2 + bx + c < 0 \text{ (standard forms of quadratic inequalities)}\)
• \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \text{ (quadratic formula for finding roots)}\)
• \(a(x - r_1)(x - r_2) > 0 \text{ or } a(x - r_1)(x - r_2) < 0 \text{ (factored form, where } r_1, r_2 \text{ are roots)}\)
• \(\text{For } a > 0: \text{ parabola opens upward, so } ax^2 + bx + c > 0 \text{ when } x < r_1 \text{ or } x > r_2 \text{ (if } r_1 < r_2\text{)}\)
• \(\text{For } a < 0: \text{ parabola opens downward, so } ax^2 + bx + c > 0 \text{ when } r_1 < x < r_2 \text{ (if } r_1 < r_2\text{)}\)

易错点

• ⚠️ Forgetting to reverse the inequality sign when multiplying or dividing both sides by a negative number during the solving process, which leads to incorrect solution intervals.
• ⚠️ Incorrectly determining which intervals satisfy the inequality by failing to properly test points in each region created by the roots, or misinterpreting the sign of the parabola based on the leading coefficient \(a\).
• ⚠️ Confusing the use of parentheses and brackets in interval notation—using parentheses \((a, b)\) for non-inclusive endpoints (when the inequality is strict: \(>\) or \(<\)) and brackets \([a, b]\) for inclusive endpoints (when the inequality is non-strict: \(\geq\) or \(\leq\)).
• ⚠️ Failing to consider the case where the discriminant \(b^2 - 4ac < 0\), which means the quadratic has no real roots and the entire parabola lies above or below the x-axis, making the solution either all real numbers or the empty set.