Question

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

Answer 1

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

=$x^2-2\times \left(\frac{1}{2}\right)\times (\surd 2+1)x+\left(\frac{(\surd 2+1{)}^2}{4}\right)-\left(\frac{(\surd 2+1{)}^2}{4}\right)+\surd 2$= 0

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

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

or , -$\frac{\surd 2-1}{2}$

= x = √2 , or, 1

ANS= x = √2 or 1.