The equation √(x + 4) - √(x - 3) + 1 = 0 has A) one real root�

1.	The equation √(x + 4) - √(x - 3) + 1 = 0 has A) one real root B) two imaginary roots C) no root D) one real and one imaginary root
Answer» 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. Correct option is C) no root

The equation √(x + 4) - √(x - 3) + 1 = 0 has A) one real root B) two imaginary roots C) no root D) one real and one imaginary root

Answer»

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.

Correct option is C) no root