Question

Solve (u + 2) ^ 2 - 8= 0 , where u is a real number. Simplify your answer as much as possible . If there is more than one solution, separate them with commas. If there is no solution, click "No solution."

Answer 1

The equation is given as

`(u+2)^2-8=0`

Before we begin solving, we need to expand the equation:

`\begin{gathered} (u+2)(u+2)-8=0 \\ u^2+2x+2x+4-8=0 \\ u^2+4x-4=0 \end{gathered}`

We can solve the equation using the quadratic formula:

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

where

`\begin{gathered} a=1 \\ b=4 \\ c=-4 \end{gathered}`

Substituting, we have

`\begin{gathered} u=\frac{-4\pm\sqrt[]{4^2-(4*1*-4)}}{2*1} \\ u=\frac{-4\pm\sqrt[]{32}}{2} \end{gathered}`

Therefore, we can calculate the values of u to be

`\begin{gathered} u=\frac{-4+\sqrt[]{32}}{2} \\ u=-2+2\sqrt[]{2} \end{gathered}`

or

`\begin{gathered} u=\frac{-4-\sqrt[]{32}}{2} \\ u=-2-2\sqrt[]{2} \end{gathered}`

Therefore, the roots are given as

`u=\mleft\lbrace-2+2\sqrt[]{2},-2-2\sqrt[]{2}\mright\rbrace`