The number of roots satisfying the equation √(5 - x) = x√(5 - x)�

1.	The number of roots satisfying the equation √(5 - x) = x√(5 - x) is are A) 2 B) 3 C) 1D) unlimited
Answer» Correct option is (A) 2 Given equation is \(\sqrt{5-x}=x\sqrt{5-x}\) _______________(1) \(\Rightarrow5-x=x^2(5-x)\) (By squaring both sides) \(\Rightarrow x^2(5-x)-(5-x)=0\) \(\Rightarrow(5-x)(x^2-1)=0\) \(\Rightarrow5-x=0\;or\;x^2-1=0\) \(\Rightarrow x=5\;or\;x=1\;or\;x=-1\) Put x = -1 in equation (1), we get \(\sqrt{5-(-1)}=-1\sqrt{5-(-1)}\) \(\sqrt6=-\sqrt6\) (Not satisfied) Hence, x = -1 is not a solution of given equation. But x = 1 & x = 5 are roots of given equation. \(\therefore\) Number of roots satisfying the given equation is 2. Correct option is A) 2

The number of roots satisfying the equation √(5 - x) = x√(5 - x) is are A) 2 B) 3 C) 1D) unlimited

Answer»

Correct option is (A) 2

Given equation is

\(\sqrt{5-x}=x\sqrt{5-x}\) _______________(1)

\(\Rightarrow5-x=x^2(5-x)\) (By squaring both sides)

\(\Rightarrow x^2(5-x)-(5-x)=0\)

\(\Rightarrow(5-x)(x^2-1)=0\)

\(\Rightarrow5-x=0\;or\;x^2-1=0\)

\(\Rightarrow x=5\;or\;x=1\;or\;x=-1\)

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

\(\sqrt{5-(-1)}=-1\sqrt{5-(-1)}\)

\(\sqrt6=-\sqrt6\) (Not satisfied)

Hence, x = -1 is not a solution of given equation.

But x = 1 & x = 5 are roots of given equation.

\(\therefore\) Number of roots satisfying the given equation is 2.

Correct option is A) 2