AS & A Level · Practice questions · Quadratic equations and inequalities

Roots first, then the region.

Six original Cambridge-style questions on solving quadratic equations and quadratic inequalities, from factorising to a two-condition problem in context.

Original questions All questions on this page are original work, written in the Cambridge AS & A Level style. They are not from past papers. They test the same concepts and skills the syllabus rewards.

The method

Zero on one side, roots, sketch, read.

Rearrange so one side is zero; never divide by \(x\).
Find the critical values by factorising, completing the square or the formula.
Sketch the parabola through them (make \(a > 0\) by multiplying by \(-1\) and reversing the sign).
Read: \(< 0\) between the roots, \(> 0\) outside, using \(\le\)/\(\ge\) to include the ends.
Write an outside answer as two pieces joined by 'or'.

Recall

[2 marks]

Solve \(2x^{2} - 7x + 3 = 0\).

\((2x - 1)(x - 3) = 0\). ✓
\(x = \dfrac{1}{2}\) or \(x = 3\). ✓

Calculation

[3 marks]

Solve \(5x^{2} - 3x - 1 = 0\), giving your answers correct to 3 significant figures.

\(b^{2} - 4ac = 9 + 20 = 29\). ✓
\(x = \dfrac{3 \pm \sqrt{29}}{10}\). ✓
\(x = 0.839\) or \(x = -0.239\). ✓

Calculation

[3 marks]

Solve the inequality \(x^{2} - x - 20 \ge 0\).

\((x - 5)(x + 4) = 0\) gives critical values \(5\) and \(-4\). ✓
The parabola opens upwards, so \(\ge 0\) outside the roots, ends included. ✓
\(x \le -4\) or \(x \ge 5\). ✓

Analysis

[3 marks]

Solve the inequality \((2x - 1)(x + 2) < 3x + 4\), giving your answer in exact form.

Expand: \(2x^{2} + 3x - 2 < 3x + 4\). ✓
Simplify: \(2x^{2} < 6 \Rightarrow x^{2} < 3\). ✓
\(-\sqrt{3} < x < \sqrt{3}\). ✓

Analysis

[4 marks]

The equation \(kx^{2} + 2kx + 3 = 0\), where \(k\) is a non-zero constant, has no real roots. Find the set of possible values of \(k\).

No real roots: \((2k)^{2} - 4(k)(3) < 0\). ✓
\(4k^{2} - 12k < 0 \Rightarrow 4k(k - 3) < 0\). ✓
Critical values \(0\) and \(3\); a positive quadratic in \(k\) is negative between them. ✓
\(0 < k < 3\). ✓

Analysis

[5 marks]

A rectangle has length \((x + 5)\) cm and width \((x - 1)\) cm, where \(x > 1\). Its area is at least 40 cm² and its perimeter is less than 32 cm.

Find the set of possible values of \(x\).

Area: \((x + 5)(x - 1) \ge 40 \Rightarrow x^{2} + 4x - 45 \ge 0\). ✓
\((x + 9)(x - 5) \ge 0\), so \(x \le -9\) or \(x \ge 5\); with \(x > 1\) this gives \(x \ge 5\). ✓
Perimeter: \(2\big[(x + 5) + (x - 1)\big] = 4x + 8 < 32 \Rightarrow x < 6\). ✓
Both conditions: \(5 \le x < 6\). ✓

Mark this once you have attempted all six and checked your working. It records a Practiced badge on the topic and adds a one-time bonus. Revealing the solutions alone does not count.