Correct option is (C) no root
Given equation is
\(\sqrt{x+4}-\sqrt{x-3}+1=0\) ______________(1)
\(\Rightarrow\) \(\sqrt{x+4}+1=\sqrt{x-3}\)
\(\Rightarrow x+4+1+2\sqrt{x+4}=x-3\) (By squaring both sides)
\(\Rightarrow2\sqrt{x+4}=x-3-x-5\)
\(\Rightarrow2\sqrt{x+4}=-8\)
\(\Rightarrow\sqrt{x+4}=-4\)
\(\Rightarrow x+4=(-4)^2=16\) (By squaring both sides)
\(\Rightarrow x=16-4=12\)
Put x = 12 in equation (1), we get
\(\sqrt{12+4}-\sqrt{12-3}+1=0\)
\(\Rightarrow4-3+1=0\)
\(\Rightarrow2=0\) (Not satisfy)
\(\therefore\) x = 12 is not a root of given equation.
It implies given equation has no root.
