Answer :
Answer: The solution is x = -2.
Step-by-step explanation: The given equation is as follows:
[tex]\sqrt{2x+13}-5=x.[/tex]
We will be using the following algebraic identities:
[tex](i)~(a+b)^2=a^2+2ab+b^2,\\\\(ii)~(\sqrt{a+b})^2=a+b.[/tex]
The solution is as follows:
[tex]\sqrt{2x+13}-5=x\\\\\Rightarrow \sqrt{2x+13}=x+5\\\\\Rightarrow (\sqrt{2x+13})^2=(x+5)^2\\\\\Rightarrow 2x+13=x^2+10x+25\\\\\Rightarrow x^2+8x+12=0\\\\\Rightarrow x^2+6x+2x+12=0\\\\\Rightarrow x(x+6)+2(x+6)=0\\\\\Rightarrow (x+2)(x+6)=0\\\\\Rightarrow x+2=0,~~~~~~x+6=0\\\\\Rightarrow x=-2,~-6.[/tex]
If we substitute x = -2, then
[tex]L.H.S=\sqrt{2\times (-2)+13}-5=\sqrt9-5=3-5=-2=R.H.S.[/tex]
Ie we substitute x = -6, then
[tex]L.H.S=\sqrt{2\times (-6)+13}-5=\sqrt1-5=1-5=-4\neq -6=x=R.H.S.[/tex]
Since x = -6 does not satisfy the given equation, so the solution is x = -2.
Thus, the solution is x = -2.