The answer
(a) \((3x-4y)(3x+4y)\)
(b)(i) \(\dfrac{5y^2}{9x}\)
(ii) \(\dfrac{4x+15}{(2x-3)(x+2)}\)
(c) \(x = -\dfrac12\) or \(x = 5\)
(d)(i) \(\left(x-\dfrac92\right)^2 - \dfrac{13}{4}\)
(ii) \(x = 6.30\) or \(x = 2.70\)
O-Level E-Math 2015 Paper 2 Question 1 · Verified worked solution by the Genius Plus Academy teaching team
The question
(a) Factorise \(9x^2 - 16y^2\). [1]
(b) Express as a single fraction in its simplest form (i) \(\dfrac{15xy}{12} \div \dfrac{9x^2}{4y}\), [1] (ii) \(\dfrac{6}{2x-3} - \dfrac{1}{x+2}\). [2]
(c) Solve the equation \(\dfrac{9}{x-4} = 2x - 1\). [3]
(d) (i) Express \(x^2 - 9x + 17\) in the form \((x + a)^2 + b\). [1] (ii) Hence solve \(x^2 - 9x + 17 = 0\), to two decimal places. [3]
(a) Difference of two squares: \(9x^2 - 16y^2 = (3x - 4y)(3x + 4y)\).
(b)(i) \(\dfrac{15xy}{12} \times \dfrac{4y}{9x^2} = \dfrac{60xy^2}{108x^2} = \dfrac{5y^2}{9x}\).
(ii) Common denominator \((2x-3)(x+2)\): \(\dfrac{6(x+2) - (2x-3)}{(2x-3)(x+2)} = \dfrac{6x+12-2x+3}{(2x-3)(x+2)} = \dfrac{4x+15}{(2x-3)(x+2)}\).
(c) \(9 = (2x-1)(x-4) = 2x^2 - 9x + 4\), so \(2x^2 - 9x - 5 = 0 \Rightarrow (2x+1)(x-5) = 0\), giving \(x = -\dfrac12\) or \(x = 5\).
(d)(i) \(x^2 - 9x + 17 = \left(x - \dfrac92\right)^2 - \dfrac{81}{4} + 17 = \left(x - \dfrac92\right)^2 - \dfrac{13}{4}\) (so \(a = -\dfrac92\), \(b = -\dfrac{13}{4}\)).
(ii) \(\left(x - \dfrac92\right)^2 = \dfrac{13}{4} \Rightarrow x - 4.5 = \pm\sqrt{3.25} = \pm 1.8028\), so \(x = 6.30\) or \(x = 2.70\) (2 d.p.).
Answer: (a) \((3x-4y)(3x+4y)\)
(b)(i) \(\dfrac{5y^2}{9x}\)
(ii) \(\dfrac{4x+15}{(2x-3)(x+2)}\)
(c) \(x = -\dfrac12\) or \(x = 5\)
(d)(i) \(\left(x-\dfrac92\right)^2 - \dfrac{13}{4}\)
(ii) \(x = 6.30\) or \(x = 2.70\)
Same structure, different numbers
Swap the constants, dress a quadratic as a length, hide a derivative inside an integral, and a student sees a brand new problem. The structure underneath is the same, and so is the method. Once a student can name the structure, a whole row of questions that look different start to open the same way.
That is where marks really leak: in choosing the method, not in the algebra that follows. We call it Lock and Key, name the lock, then the key follows.
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Genius Plus Academy · O-Level & IP Mathematics
Our O-Level E-Math tuition trains the same recognise-the-structure method these worked solutions show, taught by a team that has marked these papers for years. It runs within our weekly Secondary Math programme, Sec 1 to 4 and IP.
It is a difference of two squares question from Algebraic manipulation / quadratics, worth 11 marks: (a) 1, (b) 1 + 2, (c) 3, (d) 1 + 3.
Yes. IP (Integrated Programme) schools teach the same O-Level Mathematics content; they just sequence it differently and set their own internal exams, so these worked solutions apply to IP students too.
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