The equation \(\sqrt{x+4}\) - \(\sqrt{x-3}\) + 1 = 0 has .........

1.	The equation \(\sqrt{x+4}\) - \(\sqrt{x-3}\) + 1 = 0 has ..........A) no root B) one real root C) one real root and one imaginary root D) two imaginary roots
Answer» Correct option is (A) no root Given equation is \(\sqrt{x+4}-\sqrt{x-3}+1=0\) ______________(1) \(\Rightarrow\) \(\sqrt{x+4}+1=\sqrt{x-3}\) \(\Rightarrow\) \((\sqrt{x+4}+1)^2=x-3\) (By squaring both sides) \(\Rightarrow x+4+2\sqrt{x+4}+1=x-3\) \(\Rightarrow2\sqrt{x+4}=x-3-x-5\) \(\Rightarrow2\sqrt{x+4}=-8\) \(\Rightarrow\) 4 (x+4) = 64 (By squaring both sides) \(\Rightarrow\) x+4 = \(\frac{64}4\) = 16 \(\Rightarrow\) x = 16 - 4 = 12 Put x = 12 in equation (1), we get \(\sqrt{12+4}-\sqrt{12-3}+1=0\) \(\Rightarrow\) \(\sqrt{16}-\sqrt{9}+1=0\) \(\Rightarrow\) 4 - 3 + 1 = 0 \(\Rightarrow\) 2 = 0 (Not satisfied) Hence, x = 12 is not a solution of given equation. Hence, the given equation has no root. Correct option is A) no root

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

Answer»

Correct option is (A) no root

Given equation is

\(\sqrt{x+4}-\sqrt{x-3}+1=0\) ______________(1)

\(\Rightarrow\) \(\sqrt{x+4}+1=\sqrt{x-3}\)

\(\Rightarrow\) \((\sqrt{x+4}+1)^2=x-3\) (By squaring both sides)

\(\Rightarrow x+4+2\sqrt{x+4}+1=x-3\)

\(\Rightarrow2\sqrt{x+4}=x-3-x-5\)

\(\Rightarrow2\sqrt{x+4}=-8\)

\(\Rightarrow\) 4 (x+4) = 64 (By squaring both sides)

\(\Rightarrow\) x+4 = \(\frac{64}4\) = 16

\(\Rightarrow\) x = 16 - 4 = 12

Put x = 12 in equation (1), we get

\(\sqrt{12+4}-\sqrt{12-3}+1=0\)

\(\Rightarrow\) \(\sqrt{16}-\sqrt{9}+1=0\)

\(\Rightarrow\) 4 - 3 + 1 = 0

\(\Rightarrow\) 2 = 0 (Not satisfied)

Hence, x = 12 is not a solution of given equation.

Hence, the given equation has no root.

Correct option is A) no root