Question

Can you help me find the roots of the quadratic equation by completing the square?

Answer 1

Given

$x^2-8x=9$

Complete the square for x as shown below

$\begin{gathered} x^2-8x+b=(x-a)^2 \\ \Rightarrow-8x=-2ax,b=a^2 \\ \Rightarrow a=4\Rightarrow b=16 \\ \Rightarrow x^2-8x+16=(x-4)^2 \end{gathered}$

Thus,

$\begin{gathered} \Rightarrow x^2-8x+16=9+16 \\ \Rightarrow(x-4)^2=25 \end{gathered}$

After completing the square, the result is (x-4)^2=25

Then, finding the roots of the quadratic equation,

$\begin{gathered} (x-4)^2=25 \\ \Rightarrow(x-4)=\sqrt{25}=\pm5 \\ \Rightarrow x=4\pm5 \\ \Rightarrow x=-1,9 \end{gathered}$

Hence, the roots are x=-1, and x=9