Question

$x=\sqrt{x+6}$ x∈R

(please explain how I can find the answer)

Answer 1

Answer:

$\displaystyle{\boxed{\red{\sf\:x\:=\:3\:}}\sf\:\quad\:OR\:\quad\:\boxed{\red{\sf\:x\:=\:-\:2\:}}}$

Step-by-step-explanation:

The given equation is

$\displaystyle{\sf\:x\:=\:\sqrt{x\:+\:6}}$

We have to find the value of x.

Now,

$\displaystyle{\sf\:x\:=\:\sqrt{x\:+\:6}}$

By squaring both sides, we get,

$\displaystyle{\implies\sf\:x^2\:=\:x\:+\:6}$

$\displaystyle{\implies\sf\:x^2\:-\:x\:-\:6\:=\:0}$

$\displaystyle{\implies\sf\:x^2\:-\:3x\:+\:2x\:-\:6\:=\:0}$

$\displaystyle{\implies\sf\:x\:(\:x\:-\:3\:)\:+\:2\:(\:x\:-\:3\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:x\:-\:3\:)\:(\:x\:+\:2\:)\:=\:0}$

$\displaystyle{\implies\sf\:(\:x\:-\:3\:)\:=\:0\:\quad\:OR\:\quad\:(\:x\:+\:2\:)\:=\:0}$

$\displaystyle{\implies\sf\:x\:-\:3\:=\:0\:\quad\:OR\:\quad\:x\:+\:2\:=\:0}$

$\displaystyle{\implies\:\underline{\boxed{\red{\sf\:x\:=\:3\:}}}\sf\:\quad\:OR\:\quad\:\underline{\boxed{\red{\sf\:x\:=\:-\:2\:}}}}$

Answer 2

Step-by-step explanation:

${\implies}{ \boxed{\mathbb{\orange{given \: that}}}} \: \\ x = \sqrt{x + 6} \\ to \: find \: value \: of \: x \\ solution \\{\implies}{ \boxed{\mathbb{\red{ squarring \: both \: side \: we \: get}}}} \\ x {}^{2} = x + 6 \\ x {}^{2} - x - 6 = 0 \\ taking \: factor \\ {x}^{2} - 3x + 2x - 6 = 0 \\ {x}(x - 3) + 2(x - 3) = 0 \\ equating \: with \: zero \: we \: get \\ x + 2 = 0 \: or \: x - 3 = 0 \\ {\implies}{ \boxed{\mathbb{\pink{x = - 2 \: or \: x = 3}}}}$