Option C: [tex]x=0[/tex] and [tex]x=\frac{5}{2}[/tex] are the solutions.
Explanation:
The equation is [tex]\frac{x}{3} =\frac{x^{2} }{x+5}[/tex]
We shall determine the value of x, by simplifying the equation.
[tex]$\begin{aligned} x(x+5) &=3 x^{2} \\ x^{2}+5 x &=3 x^{2} \\ 2 x^{2}-5 x &=0 \\ x(2 x-5) &=0 \end{aligned}$[/tex]
Thus, [tex]x=0[/tex] and [tex]x=\frac{5}{2}[/tex] are the solutions.
Now, let us check whether the solutions are extraneous solutions.
Let us substitute [tex]x=0[/tex] in the original equation to check whether both sides of the equation are equal.
[tex]\begin{aligned}&\frac{0}{3}=\frac{0^{2}}{0+5}\\&0=\frac{0}{5}\\&0=0\end{aligned}[/tex]
Thus, both sides of the equation are equal.
Hence [tex]x=0[/tex] is a true solution.
Now, Let us substitute [tex]x=\frac{5}{2}[/tex] in the original equation to check whether both sides of the equation are equal.
[tex]\begin{aligned}\frac{\left(\frac{5}{2}\right)}{3} &=\frac{\left(\frac{5}{2}\right)^{2}}{\left(\frac{5}{2}\right)+5} \\\frac{5}{6} &=\frac{\left(\frac{25}{4}\right)}{\left(\frac{15}{2}\right)} \\\frac{5}{6} &=\frac{5}{6}\end{aligned}[/tex]
Thus, both sides of the equation are equal.
Hence, [tex]x=\frac{5}{2}[/tex] is a true solution.
Thus, solutions are not extraneous.
Hence, Option C is the correct answer.