Answer
The solution set is$\underline{\left\{ 5\pm 7i \right\}}$.
Work Step by Step
${{\left( x-5 \right)}^{2}}=-49$
Simplifying, ${{\left( a-b \right)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$
$\begin{align}
& {{x}^{2}}-2\times x\times 5+25=-49 \\
& {{x}^{2}}-10x+25=-49 \\
\end{align}$
Adding $49$ to both sides:
${{x}^{2}}-10x+25+49=-49+49$
Simplifying:
${{x}^{2}}-10x+74=0$
As the equation is in $a{{x}^{2}}+bx+c=0$ form.
Applying quadratic formulae here $x=\frac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$.
Coefficients are $a=1,b=-10,c=74$ .
Substituting in the formula:
$x=\frac{10\pm \sqrt{{{\left( -10 \right)}^{2}}-4\times 1\times 74}}{2}$
$\begin{align}
& x=\frac{10\pm \sqrt{100-296}}{2} \\
& x=\frac{10\pm \sqrt{-196}}{2} \\
& x=\frac{10\pm \sqrt{{{\left( 14i \right)}^{2}}}}{2} \\
& =\frac{10\pm 14i}{2}
\end{align}$
Simplifying:
$\begin{align}
& x=\frac{10}{2}\pm \frac{14i}{2} \\
& x=5\pm 7i \\
\end{align}$
Hence, the solution is $x=5\pm 7i$.
Hence, solution set is$\left\{ 5\pm 7i \right\}$.
