Answer
no solution
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given radical equation, $
\sqrt{x^2-15x+15}=x-5
,$ square both sides of the equation. Use special products and concepts of solving quadratic equations. Finally, do checking of the solution with the original equation.
$\bf{\text{Solution Details:}}$
Squaring both sides of the equation results to
\begin{array}{l}\require{cancel}
\left( \sqrt{x^2-15x+15} \right)^2=(x-5)^2
\\\\
x^2-15x+15=(x-5)^2
.\end{array}
Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to
\begin{array}{l}\require{cancel}
x^2-15x+15=(x)^2-2(x)(5)+(5)^2
\\\\
x^2-15x+15=x^2-10x+25
.\end{array}
Using the properties of equality to isolate the variable results to
\begin{array}{l}\require{cancel}
(x^2-x^2)+(-15x+10x)=25-15
\\\\
-5x=10
\\\\
x=\dfrac{10}{-5}
\\\\
x=-2
.\end{array}
Upon checking, $
x=-2
$ DOES NOT satisfy the original equation. Hence, there is $\text{
no solution
.}$
