Quadratic inequalities form a critical component in A-Level Mathematics, offering valuable insights into the range of solutions for quadratic expressions. This section covers techniques for solving quadratic inequalities, with a focus on graphical methods and test points. We also introduce the concept of expressing solutions using interval notation, providing a clear and concise method for communicating mathematical results.

**Techniques for Solving Quadratic Inequalities**

**Case 1: Distinct Roots**

Consider inequalities where d < \beta:

(x-d)(x-\beta)<0 \Longrightarrow d < x < \beta (x-d)(x-\beta)>0 \Longrightarrow x < d \ or \ x > \beta.This case demonstrates inequalities with distinct roots, allowing a straightforward identification of solution intervals.

**Case 2: No Linear Term**

When the quadratic lacks a linear $( x )$ term:

x^2 - c > 0 \Longrightarrow x < -\sqrt{c} \text{ or } x > \sqrt{c},$x^2 - c \leq 0 \Longrightarrow -\sqrt{c} \leq x \leq \sqrt{c}.$These are quadratic expressions without a linear component, simplifying the solution process.

**Graphical Interpretations**

Graphical interpretation involves plotting the quadratic equation's corresponding parabola. This visual aid is instrumental in identifying the regions where the inequality holds true.

**Example 1**

Consider:

x^2 + 6x + 8 < 0.Steps to solve:

1. Factorise: $x^2 + 6x + 8 = (x + 4)(x + 2).$

2. Identify the roots: $x = -4$ and $x = -2.$

3. Sketch the parabola, intersecting the x-axis at these roots.

4. The solution is where the parabola is below the x-axis.

Thus, the solution is -4 < x < -2, or in interval notation, $(-4, -2)$.

**Example 2**

Solve the inequality:

x^2 - 5x + 6 > 0.Steps to solve:

1. Factorise: $x^2 - 5x + 6 = (x - 2)(x - 3)$.

2. Find roots: $x = 2, x = 3$.

3. Draw the parabola, intersecting the x-axis at $x = 2$ and $x = 3$.

4. The solution lies where the parabola is above the x-axis.

Therefore, the solution is x < 2 or x > 3, in interval notation: $(-\infty, 2) \cup (3, \infty)$.

**Example 3**

Consider:

$-x^2 + 4x - 3 \leq 0.$Steps to solve:

1. Rearrange: $x^2 - 4x + 3 \geq 0$.

2. Factorise: $x^2 - 4x + 3 = (x - 1)(x - 3)$.

3. Determine roots: $x = 1, x = 3$.

4. Sketch the parabola, noting intersections at the roots.

The solution, where the parabola is above or on the x-axis, is $x \leq 1$ or $x \geq 3$, or in interval notation: $(-\infty, 1] \cup [3, \infty).$

**Interval Notation**

Interval notation is a concise way to express ranges of solutions. This format is especially useful in clearly defining the sets of solutions for quadratic inequalities.

Rahil spent ten years working as private tutor, teaching students for GCSEs, A-Levels, and university admissions. During his PhD he published papers on modelling infectious disease epidemics and was a tutor to undergraduate and masters students for mathematics courses.