Question

Solve ${\log }_{2}x+\frac{1}{2}{\log }_2(x+2)=2$

Answer 1

${\log }_{2}x+\frac{1}{2}{\log }_2\left(x+2\right)=2$

$\Rightarrow {\log }_{2}x+{\log }_2\sqrt{x+2}=2$

$\Rightarrow {\log }_{2}x\sqrt{x+2}=2$

$\Rightarrow x\sqrt{x+2}=2^2=4$

$\Rightarrow x^2\sqrt{x+2}=4$

$\Rightarrow x^2\left(x+2\right)=16$

$\Rightarrow x^3+2x^2-16=0$

$\Rightarrow \left(x-2\right)\left(x^2+4x+8\right)=0$

$\Rightarrow \left(x-2\right)=0\left[\because x^2+4x+8\right]$ [has imaginary zeroes]

$\Rightarrow x=2$