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Quadratic and Fractional Equations

Secondary 3 practice

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Quadratic and Fractional Equations practice questions

Secondary 3 topics from the O-Level Mathematics 4052 syllabus. Every question below has a full worked solution.

When a student gets a question wrong in Math Amigo, it does not show the answer straight away. It reads the working, identifies the step that went wrong, and asks a question that points at the correction. The worked solution is there once a genuine attempt has been made.

Practise Quadratic and Fractional Equations free

What this topic covers

Worked examples

Example 1

Solve x2−8x+3=0x^2 - 8x + 3 = 0 by completing the square, giving your answers correct to 2 decimal places. State both values of xx.
Show the worked solution
  • x2−8x+3=0x^2 - 8x + 3 = 0
  • x2−8x+16−16+3=0add and subtract (82)2=16x^2 - 8x + 16 - 16 + 3 = 0 \quad \scriptsize\textit{add and subtract }\left(\frac{8}{2}\right)^2 = 16
  • (x−4)2−13=0(x - 4)^2 - 13 = 0
  • (x−4)2=13(x - 4)^2 = 13
  • x−4=±13take square roots on both sidesx - 4 = \pm\sqrt{13} \quad \scriptsize\textit{take square roots on both sides}
  • x=4±13x = 4 \pm \sqrt{13}
  • x=7.61 or x=0.39(to 2 d.p.)x = 7.61 \text{ or } x = 0.39 \quad \text{(to 2 d.p.)}
  • ∴the larger value of x=7.61 and the smaller value of x=0.39\therefore \text{the larger value of } x = 7.61 \text{ and the smaller value of } x = 0.39

Answer: larger=7.61,smaller=0.39larger=7.61,smaller=0.39

Example 2

Solve 3x2−x=73x^2 - x = 7, giving your answers correct to 3 significant figures. State both values of xx.
Show the worked solution
  • 3x2−x−7=0rearrange3x^2 - x - 7 = 0 \quad \scriptsize\textit{rearrange}
  • Identify: a=3a = 3, b=−1b = -1, c=−7c = -7
  • x=−b±b2−4ac2ax = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}
  • x=1±(−1)2−4(3)(−7)6x = \dfrac{1 \pm \sqrt{(-1)^2 - 4(3)(-7)}}{6}
  • x=1±856x = \dfrac{1 \pm \sqrt{85}}{6}
  • x=1+856orx=1−856x = \dfrac{1 + \sqrt{85}}{6} \quad \text{or} \quad x = \dfrac{1 - \sqrt{85}}{6}
  • x=1.70x = 1.70 or x=−1.37x = -1.37 (to 3 s.f.)
  • The larger value of xx is 1.701.70 and the smaller value of xx is −1.37-1.37.

Answer: larger=1.70,smaller=−1.37larger=1.70,smaller=-1.37

Example 3

Express y=x2−2x+7y = x^2 - 2x + 7 in the form y=(x−p)2+qy = (x - p)^2 + q. State the value of qq.
Show the worked solution
  • y=x2−2x+7y = x^2 - 2x + 7
  • y=x2−2x+1−1+7add and subtract (2/2)2y = x^2 - 2x + 1 - 1 + 7 \quad \scriptsize\textit{add and subtract }(2/2)^2
  • y=(x−1)2+6y = (x - 1)^2 + 6
  • The turning point is (1, 6)(1,\, 6), so q=6q = 6.

Answer: 66

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Which syllabus does this follow?

O-Level Mathematics (4052), for Singapore secondary students. Math Amigo covers Secondary 1 to 4, both O-Level Mathematics and Additional Mathematics.

Will it just show my child the answer?

No. On a wrong answer it looks at the working, points to the step that went wrong, and asks a question that helps the student find the correction. The full worked solution is available after a genuine attempt.

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