Practice › Number and Algebra › Quadratic Inequalities

Quadratic Inequalities: Paper 1 Long Questions

Every quadratic inequality comes down to one picture: a parabola and the x-axis. These ten questions start with factorising and reading the sign off a sketch, then move to discriminant conditions, completing the square and word problems.

10 questions · 60 marks · about 60 minutes · answers below each question

All questions are original, written for Math and AI Academy and modelled on past-paper style. They are not reproductions of HKEAA examination questions.

Question 1 · Original — extension topic, no past-paper model

Tests: solving a quadratic inequality by factorising; the closed and open forms; counting integers.

(a) Solve $x^{2} - x - 12 < 0$ .(2)

(b) Solve $x^{2} - x - 12 \geq 0$ .(1)

(c) Write down the number of integers satisfying $x^{2} - x - 12 < 0$ .(1)

Show answer

(a) $- 3 < x < 4$ . The graph opens upwards, so it is below the x-axis between the roots $- 3$ and $4$ .

(b) $x \leq - 3$ or $x \geq 4$ . This is the rest of the number line, now including the roots.

(c) 6. The integers are $- 2, - 1, 0, 1, 2, 3$ .

Question 2 · Original — extension topic, no past-paper model

Tests: quadratic with leading coefficient other than 1; counting integers.

(a) Solve $3 x^{2} - 5 x - 2 \leq 0$ .(3)

(b) Write down the number of integers satisfying $3 x^{2} - 5 x - 2 \leq 0$ .(1)

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(a) $- \frac{1}{3} \leq x \leq 2$ . It factorises as $(3 x + 1) (x - 2) \leq 0$ .

(b) 3. The integers are $0, 1, 2$ ; $- 1$ lies below $- \frac{1}{3}$ .

Question 3 · Original — extension topic, no past-paper model

Tests: rearranging to standard form; why you must not divide by x.

(a) Solve $(x + 2) (x - 3) > 6$ .(3)

(b) Solve $x^{2} \geq 9 x$ .(3)

Show answer

(a) $x < - 3$ or $x > 4$ . Expand and move 6 across to get $x^{2} - x - 12 > 0$ .

(b) $x \leq 0$ or $x \geq 9$ . Move $9 x$ across and factorise as $x (x - 9) \geq 0$ . Dividing by $x$ would lose the branch $x \leq 0$ .

Question 4 · Original — extension topic, no past-paper model

Tests: perfect squares; showing an expression is always positive by completing the square.

(a) Solve $x^{2} - 6 x + 9 > 0$ .(2)

(b) Solve $x^{2} - 6 x + 9 \leq 0$ .(2)

(c) Show that $x^{2} - 6 x + 10 > 0$ for all real values of $x$ .(2)

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(a) All real $x$ except $x = 3$ . The expression is $(x - 3)^{2}$ , which is positive unless $x = 3$ .

(b) $x = 3$ only. A square cannot be negative, so equality is the only possibility.

(c) $x^{2} - 6 x + 10 = (x - 3)^{2} + 1 \geq 1 > 0$ for every real $x$ .

Question 5 · Original — extension topic, no past-paper model

Tests: compound inequality: intersect a quadratic and a linear solution set.

(a) Solve the compound inequality $x^{2} - 2 x - 3 > 0$ and $3 x + 2 < 17$ .(4)

(b) Write down the greatest integer satisfying both inequalities.(1)

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(a) $x < - 1$ or $3 < x < 5$ . The quadratic gives $x < - 1$ or $x > 3$ ; the linear one gives $x < 5$ .

(b) 4. The integers in $3 < x < 5$ are just 4; 5 is excluded.

Question 6 · Original — extension topic, no past-paper model

Tests: finding a quadratic from its graph; solving f(x) greater than a number that is not on the figure.

The figure shows the graph of

y = f (x)

, where

f (x)

is a quadratic function. The graph cuts the x-axis at

A (- 2, 0)

and

B (4, 0)

, and the y-axis at

C (0, - 8)

(a) Find $f (x)$ .(3)

(b) Using the graph, solve $f (x) < 0$ .(1)

(c) Solve $f (x) > 7$ .(3)

Show answer

(a) $f (x) = x^{2} - 2 x - 8$ . Write $f (x) = a (x + 2) (x - 4)$ and use $f (0) = - 8$ to get $a = 1$ .

(b) $- 2 < x < 4$ . The graph is below the x-axis between A and B.

(c) $x < - 3$ or $x > 5$ . Solve $x^{2} - 2 x - 15 > 0$ . The line $y = 7$ is not drawn, so this part needs algebra.

Question 7 · Original — extension topic, no past-paper model

Tests: negative leading coefficient; completing the square; surd end-points.

Let

f (x) = - x^{2} + 4 x + 5

(a) Solve $f (x) \geq 0$ .(3)

(b) Express $f (x)$ in the form $- (x - h)^{2} + k$ . Hence write down the greatest value of $f (x)$ .(2)

(c) Solve $f (x) \geq 3$ . Leave your answer in surd form.(3)

Show answer

(a) $- 1 \leq x \leq 5$ . Multiplying by $- 1$ reverses the sign: $x^{2} - 4 x - 5 \leq 0$ .

(b) $f (x) = - (x - 2)^{2} + 9$ ; the greatest value is 9.

(c) $2 - 6 \leq x \leq 2 + 6$ . Use (b): $(x - 2)^{2} \leq 6$ .

Question 8 · Original — extension topic, no past-paper model

Tests: discriminant condition that leads to a quadratic inequality in the parameter.

The quadratic equation

x^{2} + k x + (k + 3) = 0

, where

k

is a constant, has no real roots.

(a) Find the range of values of $k$ .(4)

(b) Write down the number of integer values of $k$ satisfying (a).(1)

Show answer

(a) $- 2 < k < 6$ . No real roots means $Δ = k^{2} - 4 k - 12 < 0$ .

(b) 7. The integers are $- 1, 0, 1, 2, 3, 4, 5$ .

Question 9 · Original — extension topic, no past-paper model

Tests: translating a word problem into a quadratic inequality; combining it with a linear one.

A rectangular card is 5 cm longer than it is wide. Let the width of the card be

x

cm.

(a) The area of the card is at most $84 cm^{2}$ . Show that $0 < x \leq 7$ .(3)

(b) The perimeter of the card is at least $20$ cm. Find the range of values of $x$ .(3)

Show answer

(a) Area $= x (x + 5) \leq 84$ , so $(x + 12) (x - 7) \leq 0$ , giving $- 12 \leq x \leq 7$ . A width must be positive, so $0 < x \leq 7$ .

(b) $\frac{5}{2} \leq x \leq 7$ . The perimeter condition gives $4 x + 10 \geq 20$ , so $x \geq \frac{5}{2}$ .

Question 10 · Original — extension topic, no past-paper model

Tests: profit model: break-even range, greatest value by completing the square, then a surd-boundary count.

A shop sells

x

cups of juice per day. Its daily profit is

$ P

, where

P = - 2 x^{2} + 60 x - 288

and

x

is a whole number.

(a) Find the range of values of $x$ for which the shop makes a profit.(3)

(b) Find the greatest daily profit.(2)

(c) The shop wants a daily profit of at least $$126$ . How many different values of $x$ satisfy this?(4)

Show answer

(a) $6 < x < 24$ . Solve $x^{2} - 30 x + 144 < 0$ .

(b) $162, when $x = 15$ . Write $P = - 2 (x - 15)^{2} + 162$ .

(c) 9 values. $(x - 15)^{2} \leq 18$ gives $15 - 32 \leq x \leq 15 + 32$ , that is about 10.76 to 19.24, so $x = 11, \dots, 19$ .

Continue practising this topic: Try the Paper 1 Challenge for harder variants, or move to Paper 2 multiple-choice — also available as Paper 2 Challenge MC.

Where this sits in the syllabus

We found no HKDSE Compulsory Part past-paper question on quadratic inequalities between 2012 and 2026, so these sets are an extension of the exam-style practice on this site rather than a set modelled on a particular paper. The skills underneath are exactly the ones the exam does test: factorising, completing the square, the discriminant and reading a graph.

Full coverage of this topic, and the rest of the course, is on the Mathematics page. More sets are listed on the practice index.

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