Understanding Quadratic Inequalities: A Complete Student Guide

A quadratic inequality is an expression of the form $ax^2 + bx + c \ \diamond\ 0$, where $a, b, c \in \mathbb{R}$, $a \neq 0$, and $\diamond$ represents one of $>, \geq, <, \leq$. The solution set comprises all real $x$ for which the inequality holds.

Standard Forms and Notation for Quadratic Inequalities

A quadratic inequality is conventionally written as $ax^2 + bx + c \ \diamond\ 0$, requiring all terms to be on one side and zero on the other. The sign of the coefficient $a$ determines the orientation of the corresponding parabola.

If $a > 0$, the parabola opens upwards; if $a < 0$, the parabola opens downwards. Real roots occur if and only if the discriminant $D = b^2 - 4ac \geq 0$. The nature of the solution set depends on the sign of $a$, the type of inequality, and the number of real roots.

Classification of Solution Sets Based on Discriminant

Result: If $ax^2 + bx + c$ has discriminant $D = b^2 - 4ac$, the solution set of $ax^2 + bx + c \ \diamond\ 0$ is classified by:

• $D > 0$: Two distinct real roots; solution set is an interval or a union of two intervals.
• $D = 0$: One real repeated root; solution set reduces to a singleton or the entire real line, depending on the inequality.
• $D < 0$: No real roots; the solution set is either $\mathbb{R}$ or $\varnothing$ based on the sign of $a$ and the symbol $\diamond$.

For detailed structure of quadratic roots, refer to Quadratic Equations Roots.

Analysing Quadratic Inequalities Using the Sign Chart Method

To solve $ax^2 + bx + c \ \diamond\ 0$, first factor or use the quadratic formula to find real roots $\alpha$ and $\beta$ ($\alpha \leq \beta$). The real axis is partitioned at these roots, and the sign of $ax^2 + bx + c$ alternates in each interval.

For $a > 0$, the function is positive for $x < \alpha$ and $x > \beta$, and negative between the roots. For $a < 0$, these signs are reversed. For strict inequalities, endpoints are excluded; for non-strict, included.

A frequent error is neglecting to reverse the inequality when multiplying through by a negative coefficient. This must be attended to in all algebraic manipulations.

Evaluation of Solution Regions with Example Cases

Consider the inequality $x^2 - 5x + 6 \leq 0$:

Given: $x^2 - 5x + 6 \leq 0$

Factor: $x^2 - 5x + 6 = (x - 2)(x - 3)$

Roots: $x = 2$, $x = 3$

Test intervals: For $x < 2$, $(x - 2)(x - 3) > 0$; for $2 \leq x \leq 3$, $(x - 2)(x - 3) \leq 0$; for $x > 3$, $(x - 2)(x - 3) > 0$

Solution: The solution set is $2 \leq x \leq 3$

For inequalities where the discriminant is negative, such as $x^2 + x + 1 < 0$, since $D < 0$ and $a > 0$, the parabola lies entirely above the $x$-axis, so no real $x$ satisfies the inequality.

Solving Quadratic Inequalities Using the Quadratic Formula

If the quadratic does not factorize easily, apply the quadratic formula:

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Real roots determine the interval bounds. The intervals satisfying the inequality are selected based on sign analysis and test points.

For expressions with rational roots or when instructed, solutions may be reported in surd or decimal form, as appropriate for the exam.

Worked Problems on Quadratic Inequality Solution Sets

Example: Solve $2x^2 - 7x + 3 > 0$

Quadratic formula yields $x = \dfrac{7 \pm \sqrt{49 - 24}}{4} = \dfrac{7 \pm 5}{4}$, so $x = 3$ or $x = \dfrac{1}{2}$.

Intervals: $(-\infty, \frac{1}{2}),\ (\frac{1}{2}, 3),\ (3, \infty)$; Test $x = 0;\ 2(0)^2 - 7(0) + 3 = 3 > 0$; $x = 2;\ 2(4) - 14 + 3 = -3 < 0$; $x = 4;\ 2(16) - 28 + 3 = 3 > 0$.

Solution: $x < \dfrac{1}{2}$ or $x > 3$

Example: Find the set of integers satisfying $x^2 - 10x + 21 \leq 0$

Factor: $(x - 3)(x - 7) \leq 0$; Roots $x = 3$, $x = 7$

Solution: $3 \leq x \leq 7$, so possible integer values are $x = 3, 4, 5, 6, 7$

Example: Solve $x^2 + 4x + 4 > 0$

Factor: $(x + 2)^2 > 0$; Repeated root at $x = -2$

Since the square is always non-negative and equals zero only at $x = -2$, the solution is $x \in \mathbb{R} \setminus \{-2\}$

Graphical Interpretation of Quadratic Inequality Solution Sets

On the Cartesian plane, solutions to $f(x) \diamond 0$ correspond to $x$ intervals where the quadratic curve lies above or below the $x$-axis. Intervals are marked by the roots and the sign of $a$. Strict inequalities use open endpoints; non-strict, closed endpoints.

On a number line, represent solution intervals with open or closed circles, as appropriate, and shaded regions indicating the set of valid $x$ values. Questions often require explicit graphical representation in exams.

Errors and Misconceptions in Quadratic Inequality Solutions

Common Error: Failing to reverse the inequality when multiplying or dividing by a negative number.

Another error is omitting the equality case in non-strict inequalities. Checking intervals and test values systematically avoids such mistakes.