Question

Solve the quadratic equation by completing the square
`{x}^(2) + 12x + 2 = 0`

Answer 1

`ax^2+bx+c=0`

To complete the square you add and subtract the next to the equation:

`((b)/(2))^2`

For the given equation:

`x^2+12x+2=0`
`((b)/(2))^2=((12)/(2))^2=6^2`

Add and subtract 6 squared as follow:

`x^2+12x+6^2+2-6^2=0`

Simplify:

`\begin{gathered} (a^2+2ab+b^2)=(a+b)^2 \\ \\ x^2+12x+6^2=(x+6)^2 \\ \\ \\ \\ (x+6)^2+2-6^2=0 \\ (x+6)^2+2-36=0 \\ (x+6)^2-34=0 \end{gathered}`

Solve x by taking square roots:

`\begin{gathered} (x+6)^2-34+34=0+34 \\ (x+6)^2=34 \\ \\ \sqrt[]{(x+6)^2}=\sqrt[]{34} \\ \\ x+6=\pm\sqrt[]{34} \\ x+6-6=\pm\sqrt[]{34}-6 \\ \\ x_1=\sqrt[]{34}-6 \\ x_2=-\sqrt[]{34}-6 \end{gathered}`Then, the solutions for the given quadratic equation are:
`\begin{gathered} x_1=\sqrt[]{34}-6 \\ x_2=-\sqrt[]{34}-6 \end{gathered}`