Question

Solve each of the following equations by using the method of completing the square:

$x^2-4x+1=0$

Answer 1

$$\begin{gathered} x^2-4x+1=0 \\ \Rightarrow x^2-4x=-1 \\ \Rightarrow x^2-2\times x\times 2+2^2=-1+2^2\left(\mathrm{Adding}{2}^2\mathrm{on}\mathrm{both}\mathrm{sides}\right) \\ \Rightarrow {\left(x-2\right)}^2=-1+4=3 \end{gathered}$$
$$\begin{gathered} \Rightarrow x-2=\pm \sqrt{3}\left(\mathrm{Taking}\mathrm{square}\mathrm{root}\mathrm{on}\mathrm{both}\mathrm{sides}\right) \\ \Rightarrow x-2=\sqrt{3}\mathrm{or}x-2=-\sqrt{3} \\ \Rightarrow x=2+\sqrt{3}\mathrm{or}x=2-\sqrt{3} \end{gathered}$$
Hence, $2+\sqrt{3}$ and $2-\sqrt{3}$ are the roots of the given equation.