Solving Quadratic Inequalities

• You find critical values by temporarily treating the inequality as an equation ($ax^2 + bx + c = 0$) and factorising it.
• Edexcel examiners expect you to confidently factorise simple quadratics, harder quadratics (where $a > 1$), expressions with a common factor, and the difference of two squares.
• For example, the inequality $x^2 - 16 < 0$ factorises to $(x - 4)(x + 4) < 0$, giving critical values of $x = 4$ and $x = -4$.

Sketching to Identify the Solution Region

Every time you draw a quick curve, you unlock the visual secret to the algebra. A quick sketch of the parabola is the most reliable way to identify the correct solution region.

• If your $x^2$ coefficient is positive, the graph will be U-shaped (concave up).
• For inequalities where the expression is $> 0$ or $\geq 0$, you need the regions above the $x$-axis, which are the outer "tails" of the U-shape.
• If the expression is $< 0$ or $\leq 0$, you need the region below the $x$-axis, which is the central "valley" between the roots.
• Your sketch does not need to be perfectly to scale, but it must clearly show the $x$-axis and correctly labelled critical values.

Writing the Final Solution

Writing your perfectly calculated answer in the wrong format can cost you the final mark. Edexcel mark schemes require specific logic patterns depending on where your region lies.

• If your solution is the valley between the roots, use a single continuous double inequality (e.g., $-2 < x < 5$).
• If your solution is the outer tails, you have disjoint inequalities, which must be written as two separate ranges joined by the word "or" (e.g., $x < -2$ or $x > 5$).
• For the Edexcel Higher Tier, you are expected to understand and apply formal set notation.
• A single continuous range is written as $\{x : -2 < x < 5\}$, while disjoint ranges use the union symbol ($\cup$), written as $\{x : x < -2\} \cup \{x : x > 5\}$.

Worked Example

Solve the quadratic inequality $x^2 - x \geq 12$.

Step 1: Rearrange the inequality so that one side is zero.

• $x^2 - x - 12 \geq 0$

Step 2: Factorise the quadratic to find the critical values.

• $(x - 4)(x + 3) = 0$
• The critical values are $x = 4$ and $x = -3$.

Step 3: Sketch the parabola to identify the required region.

• Draw a U-shaped curve crossing the $x$-axis at $-3$ and $4$.
• Because the inequality is $\geq 0$, identify the regions above the $x$-axis (the outer tails).

Step 4: State the final solution range using inequality notation.

• The solution falls outside the roots, requiring the word "or".
• $x \leq -3$ or $x \geq 4$

Step 5: State the equivalent set notation (Higher Tier).

• $\{x : x \leq -3\} \cup \{x : x \geq 4\}$