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Quadratic and Simultaneous Equations — Paper 1 Long Questions

Twelve conventional-response questions in HKDSE Paper 1 style, ordered from a plain factorisation up to a proof that an equation has real roots for every value of a constant. Mark allocations follow the real paper, so the set doubles as a timing exercise.

12 questions · about 70 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.

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Question 1 — factorising a quadratic with a leading coefficient
Question 2 — rearranging to standard form; not dividing away a root
Question 3 — substitution into a non-linear equation; pairing the solutions
Question 4 — setting up partial variation; solving the resulting quadratic
Question 5 — discriminant equal to zero; remembering both square roots
Question 6 — relations between roots and coefficients
Question 7 — forming a quadratic from a geometric context; rejecting an inadmissible root
Question 8 — discriminant inequality; the quadratic inequality it produces
Question 9 — eliminating y; discriminant as a count of solutions
Question 10 — non-real roots; building a new equation from shifted roots
Question 11 — forming an equation from times; rejecting a negative root
Question 12 — proving real roots for all k; both cases of a ratio condition

Question 1 · Modelled on 2013 P2 Q6 — recast as a conventional-response opener

Solve the equation $3 x^{2} + 5 x - 2 = 0$ . (3 marks)

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$x = \frac{1}{3}$ or $x = - 2$ . The left side factorises as $(3 x - 1) (x + 2)$ ; most marks lost here come from mis-splitting the middle term.

Question 2 · Modelled on 2014 P2 Q4 — expand first, then recognise a common factor

Solve the equation $(2 x - 3)^{2} = 4 x + 9$ . (4 marks)

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$x = 0$ or $x = 4$ . After expanding, the constant terms cancel and the equation becomes $4 x^{2} - 16 x = 0$ ; dividing by $x$ at that point would lose the root $x = 0$ .

Question 3 · Modelled on 2015 P2 Q34 — a linear equation paired with a non-linear one

Solve the simultaneous equations ${y = x + 3 x y = 10$ (4 marks)

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$x = 2, y = 5$ or $x = - 5, y = - 2$ . Substituting the linear equation into the second gives $x^{2} + 3 x - 10 = 0$ . Each value of x must be paired with its own y.

Question 4 · Modelled on 2016 P1 Q8 and 2019 P1 Q10 — partial variation leading to a quadratic

It is given that $f (x)$ is the sum of two parts, one part varying as $x$ and the other part varying as $x^{2}$ . Suppose that $f (2) = 14$ and $f (- 3) = 9$ .

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

(b) Solve the equation $f (x) = 0$ . (2 marks)

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(a) $f (x) = 2 x^{2} + 3 x$ . (b) $x = 0$ or $x = - \frac{3}{2}$ . Part (a) is a pair of simultaneous linear equations in the two constants; part (b) needs only a common factor of x.

Question 5 · Modelled on 2016 P2 Q8 — equal roots, written as a conventional-response question

Let $k$ be a real constant. The quadratic equation $x^{2} + (k + 2) x + 9 = 0$ has equal roots.

(a) Find the value(s) of $k$ . (3 marks)

(b) For each value of $k$ found in (a), solve the equation. (2 marks)

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(a) $k = 4$ or $k = - 8$ . (b) When $k = 4$ , $x = - 3$ ; when $k = - 8$ , $x = 3$ . Setting the discriminant to zero gives $(k + 2)^{2} = 36$ , which has two solutions — dropping the negative one is the usual slip.

Question 6 · Original — sum and product of roots without solving

The roots of the equation $2 x^{2} - 6 x + 5 = 0$ are $α$ and $β$ . Without solving the equation, find

(a) $α + β$ and $α β$ ; (2 marks)

(b) $α^{2} + β^{2}$ ; (2 marks)

(c) $\frac{1}{α} + \frac{1}{β}$ . (2 marks)

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(a) $α + β = 3$ , $α β = \frac{5}{2}$ . (b) $4$ . (c) $\frac{6}{5}$ . Part (b) uses $α^{2} + β^{2} = (α + β)^{2} - 2 α β$ ; part (c) is just the sum over the product.

Question 7 · Original — a word problem where one root must be rejected

The length of a rectangular garden is 4 m longer than its width, and its area is 96 m².

(a) Find the length and the width of the garden. (3 marks)

(b) A path of uniform width $t$ m is laid inside the garden along all four sides, leaving a rectangular lawn of area 45 m² in the middle. Find $t$ . (4 marks)

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(a) Width 8 m, length 12 m. (b) $t = 1.5$ . Part (b) gives $4 t^{2} - 40 t + 51 = 0$ , whose roots are 1.5 and 8.5; the second is rejected because the lawn's width $8 - 2 t$ would be negative.

Question 8 · Modelled on 2022 P1 Q10(b)(ii) — range of a constant for two distinct real roots

Let $k$ be a real constant and let $p (x) = x^{2} - 2 k x + (k + 6)$ .

(a) Find the range of values of $k$ such that the equation $p (x) = 0$ has two distinct real roots. (4 marks)

(b) Suppose instead that $p (x) = 0$ has equal roots and $k > 0$ . Solve the equation $p (x) = 0$ . (3 marks)

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(a) $k < - 2$ or $k > 3$ . (b) $k = 3$ , and the equation becomes $x^{2} - 6 x + 9 = 0$ , so $x = 3$ . The discriminant condition reduces to $(k - 3) (k + 2) > 0$ — solve it as an inequality, not by testing values.

Question 9 · Original — a line meeting a curve, handled purely as a system of equations

Consider the simultaneous equations ${y = 2 x + k y = x^{2} + 6 x + 13$ where $k$ is a real constant.

(a) Show that $x^{2} + 4 x + (13 - k) = 0$ . (1 mark)

(b) Find the range of values of $k$ such that the simultaneous equations have two distinct pairs of real solutions. (3 marks)

(c) Solve the simultaneous equations when $k = 9$ . (3 marks)

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(b) $k > 9$ . (c) $x = - 2, y = 5$ . In (b) the discriminant of the equation from (a) must be positive; in (c) the discriminant is exactly zero, so there is a single repeated solution.

Question 10 · Modelled on 2012 P2 Q34 — non-real roots and the relations that survive them

Let $α$ and $β$ be the roots of the equation $x^{2} - 6 x + 25 = 0$ .

(a) Solve the equation, giving the answers in the form $a + bi$ . (4 marks)

(b) Write down $α + β$ and $α β$ , and check them against the coefficients of the equation. (2 marks)

(c) Find the quadratic equation with integral coefficients whose roots are $α + 1$ and $β + 1$ . (3 marks)

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(a) $x = 3 + 4 i$ or $x = 3 - 4 i$ . (b) $α + β = 6$ , $α β = 25$ . (c) $x^{2} - 8 x + 32 = 0$ . The new roots have sum $α + β + 2 = 8$ and product $α β + α + β + 1 = 32$ , so (c) never needs the imaginary parts.

Question 11 · Original — a rate problem in the style of Paper 1 Section A(2)

A cyclist travels 60 km from town A to town B at a constant speed of $v$ km/h. On the return journey his constant speed is 5 km/h slower, and the return journey takes 1 hour longer than the outward journey.

(a) Find $v$ . (6 marks)

(b) Hence find the total time taken for the whole journey. (2 marks)

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(a) $v = 20$ . (b) 7 hours. The time equation clears to $v^{2} - 5 v - 300 = 0$ , giving $v = 20$ or $v = - 15$ ; a speed cannot be negative, so the second root is rejected.

Question 12 · Modelled on 2016 P1 Q14(b) and 2019 P1 Q11(b) — a discriminant that is a perfect square

Let $k$ be a real constant. Consider the equation $x^{2} + (k - 4) x + (4 - 2 k) = 0 \dots\dots (*)$

(a) Show that $(*)$ has real roots for every real value of $k$ . (3 marks)

(b) Find the value of $k$ for which $(*)$ has equal roots, and solve $(*)$ in that case. (3 marks)

(c) Express the two roots of $(*)$ in terms of $k$ . Hence find all values of $k$ for which one root of $(*)$ is three times the other. (4 marks)

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(a) The discriminant simplifies to $k^{2}$ , which is never negative. (b) $k = 0$ , and then $x = 2$ . (c) The roots are $2$ and $2 - k$ ; one is three times the other when $k = - 4$ or $k = \frac{4}{3}$ . Part (c) has two cases, because either root could be the larger one.

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

Quadratic and simultaneous equations sit in the Number and Algebra strand of the HKDSE Compulsory Part. Paper 1 tends to ask for the equation to be formed first, from a variation, a rate or a geometric setting, and only then solved. The discriminant carries most of the marks that are actually lost: it decides whether roots are two, one or non-real, and it turns questions about an unknown constant into inequalities. Everything here also feeds the next chapter, Functions and Graphs of Quadratic Functions, where the same discriminant tells you how the curve meets the x-axis.

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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