SPM Questions for Quadratic Equations

Frequently Asked Questions in SPM for Quadratic Equations

We will discuss some pass year SPM exam questions based on:

  • Solve the quadratic equations – SPM 2003, SPM 2005
  • Form a quadratic equation – SPM 2009
  • Determine the conditions for the type of roots – SPM2010

Solve the quadratic equations – SPM 2003 Paper 1 Question 3

Solve the quadratic equation 2x(x-4) =(1-x)(x+2). Give your answer correct to four significant figures.
(3 marks)

Answer:

\begin{array}{rcl}  2x(x-4) & = & (1-x)(x+2) \\  2{{x}^{2}}-8x &=& x+2-{{x}^{2}}-2x \\  2{{x}^{2}}+{{x}^{2}}-8x-x+2x &=& 2 \\  3{{x}^{2}}-7x &=& 2 \\  {{x}^{2}}-\frac{7}{3}x &=& \frac{2}{3} \\  {{x}^{2}}-\frac{7}{3}x+{{\left( -\frac{7}{3(2)} \right)}^{2}} &=& \frac{2}{3}+{{\left( -\frac{7}{3(2)} \right)}^{2}} \\  {{x}^{2}}-\frac{7}{3}x+{{\left( -\frac{7}{6} \right)}^{2}} &=& \frac{2}{3}+{{\left( -\frac{7}{6} \right)}^{2}} \\  {{\left( x-\frac{7}{6} \right)}^{2}} &=& \frac{73}{36} \\  x-\frac{7}{6} &=& \pm \sqrt{\frac{73}{36}} \\  x &=& \frac{7}{6}+\sqrt{\frac{73}{36}}\text \quad {or} \quad x = \frac{7}{6}-\sqrt{\frac{73}{36}} \\  x &=& \underline {2.59} \text \qquad {or} \quad x = \underline {- 0.2573} \\  \end{array}

Note: If you have any doubt, please refer back my previous post on how to solve the equation using completing the square method.


SPM 2005 Paper1 Question 5

Solve the quadratic equation x(2x-5) =2x-1. Give your answer correct to three decimal places.
(3 marks)

Answer:

\begin{array}{rcl}  x(2x-5)&=&2x-1 \\  2{{x}^{2}}-5x-2x+1&=&0 \\  2{{x}^{2}}-7x+1&=&0 \\  hence, a=2,b=-7,c&=&1 \\  x&=& \frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \\  x&=& \frac{-(-7)\pm \sqrt{{{(-7)}^{2}}-4(2)(1)}}{2(2)} \\  x&=& \frac{7+\sqrt{41}}{4}\text\quad{ or }\quad x=\frac{7-\sqrt{41}}{4} \\  x&=& \underline{3.351}\text\quad{ or }\quad x=\underline{0.149} \\  \end{array}


Form a quadratic equation –
SPM 2009 Paper 2 Question 2

The quadratic equation x2 – 5x + 6 = 0 has roots h and k, where h > k.

(a) Find

i) the value of h and of k.
ii) the range of x if x2 – 5x + 6 > 0
(5 marks)

(b) Using the values of h and k from 2(a)(i), form the quadratic equation which has roots h + 2 and 3k – 2.
(2 marks)

Answer:

a i) For x– 5x + 6 = 0, given roots are h and k,

ii)

(b)

h + 2
= 3 + 2
= 5

3k - 2
= 3(2) - 2
= 4

SOR = 5 + 4 = 9
\text{POR = 5 x 4 = 20}

Equations:
{{x}^{2}}-(\text{SOR)}x+(\text{POR)}=0
\underline{{{x}^{2}}-9x+20=0}


Determine the conditions for the type of roots
-
SPM 2010 Paper1 Question 5

The quadratic equation (1-p)x2 – 6x +10 = 0, where p is a constant, has two different roots. Find the range of values of p.

(3 marks)

Answer:

\begin{array}{rcl}  (1-p){{x}^{2}}-6x+10=0 \\  a=1-p,b=-6,c=10 \\  \\  \text{2 different roots, meant } \\  {{b}^{2}}-4ac>0 \\  {{(-6)}^{2}}-4(1-p)(10)>0 \\  36-40+40p>0 \\  40p>4 \\  p>\frac{4}{40} \\  \underline{p>\frac{1}{10}} \\  \end{array}

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About Cedric Low

MBA holder from University of Southern Queensland Australia (USQ) in 2009. Studied for IT and graduated from University of Hertfordshire in UK back to 2005. A system engineer turned to become a full time Maths tutor and researcher. Living in Malaysia. Editor of Perfectmaths.com

Posted on 01/08/2011, in Quadratic Equations, SPM Additional Maths and tagged , , . Bookmark the permalink. 2 Comments.

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