Question

Solve the following equation by using the method of completing the square:
$x^2-(\sqrt{2}+1)x+\sqrt{2}=0$
[2 Marks]

Answer 1

$x^2-(\sqrt{2}+1)x+\sqrt{2}=0$

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\Rightarrow$$x^2-2\times \left(\frac{1}{2}\right)\times (\surd 2+1)x+(\frac{(\surd 2+1{)}^2}{4})-(\frac{(\surd 2+1{)}^2}{4})$= - $\sqrt{2}$ [1 Mark]

<!--td {border: 1px solid #ccc;}br {mso-data-placement:same-cell;}--> $\Rightarrow$ $(x-\frac{\surd 2+1}{2}{)}^2$ = $(\frac{\surd 2+1}{2}{)}^2$ - √2

$\Rightarrow$$(x-\frac{\surd 2+1}{2}{)}^2$ = $(\frac{2+1+2\surd 2}{4}-\surd 2)$

$\Rightarrow$$(x-\frac{\surd 2+1}{2}{)}^2$ = $(\frac{3-2\surd 2}{4})$

$\Rightarrow$$(x-\frac{\surd 2+1}{2}{)}^2$ = $(\frac{(\surd 2-1{)}^2}{4})$ [0.5 Marks]

$\Rightarrow$ $(x-\frac{\surd 2+1}{2})$ = +$\frac{\surd 2-1}{2}$ or -$\frac{\surd 2-1}{2}$

$\Rightarrow$ x = √2 or 1. [0.5 Marks]