Answer
The simplified form of the expression ${{x}^{2}}-2x+2$ for $x=1+i$ is $0$.
Work Step by Step
Consider the expression, ${{x}^{2}}-2x+2$
Substitute $x=1+i$ in the expression${{x}^{2}}-2x+2$.
${{x}^{2}}-2x+2={{\left( 1+i \right)}^{2}}-2\left( 1+i \right)+2$
Use the formula ${{\left( a+b \right)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ and apply distributive law.
$\begin{align}
& {{x}^{2}}-2x+2=\left( 1+2i+{{i}^{2}} \right)-\left( 2+2i \right)+2 \\
& =1+2i+{{i}^{2}}-2-2i+2
\end{align}$
Replace the value ${{i}^{2}}=-1$.
${{x}^{2}}-2x+2=1+2i+\left( -1 \right)-2-2i+2$
Combine the real and imaginary parts.
$\begin{align}
& {{x}^{2}}-2x+2=\left( 1-1-2+2 \right)+\left( 2-2 \right)i \\
& =0
\end{align}$
Therefore, the simplified form of the expression ${{x}^{2}}-2x+2$ for $x=1+i$ is $0$.
