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 x2−4x−9=0x^2 - 4x - 9 = 0 by completing the square, giving your answers correct to 2 decimal places. State the larger value of xx.
Show the worked solution
  • $x^2 - 4x - 9 = 0$
  • $x^2 - 4x + 4 - 4 - 9 = 0 \quad \scriptsize\textit{add and subtract }\left(\frac{4}{2}\right)^2 = 4$
  • $(x - 2)^2 - 13 = 0$
  • $(x - 2)^2 = 13$
  • $x - 2 = \pm\sqrt{13} \quad \scriptsize\textit{take square roots on both sides}$
  • $x = 2 \pm \sqrt{13}$
  • $x = 5.61 \text{ or } x = -1.61 \quad \text{(to 2 d.p.)}$
  • $\therefore \text{ the larger value of } x = 5.61$

Answer: 5.615.61

Example 2

Solve 2x2+3x−5=2x2x^2 + 3x - 5 = 2x, giving your answers correct to 3 significant figures. State the larger value of xx.
Show the worked solution
  • $2x^2 + x - 5 = 0 \quad \scriptsize\textit{rearrange}$
  • Identify: $a = 2$, $b = 1$, $c = -5$
  • $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • $x = \dfrac{-1 \pm \sqrt{1^2 - 4(2)(-5)}}{4}$
  • $x = \dfrac{-1 \pm \sqrt{41}}{4}$
  • $x = \dfrac{-1 + \sqrt{41}}{4} \quad \text{or} \quad x = \dfrac{-1 - \sqrt{41}}{4}$
  • $x = 1.35$ or $x = -1.85$ (to 3 s.f.)
  • The larger value of $x$ is $1.35$.

Answer: 1.351.35

Example 3

Express y=x2+8x+15y = x^2 + 8x + 15 in the form y=(x+p)2+qy = (x + p)^2 + q. State the value of qq.
Show the worked solution
  • $y = x^2 + 8x + 15$
  • $y = x^2 + 8x + 16 - 16 + 15 \quad \scriptsize\textit{add and subtract }(8/2)^2$
  • $y = (x + 4)^2 - 1$
  • The turning point is where $x = -4$, giving $y = -1$.
  • Turning point: $(-4,\, -1)$.

Answer: −1-1

Questions parents ask

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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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