Question

The number of solutions of the equation $9x^2-18|x|+5=0$ belonging to the domain of definition of $lo{g}_e\left\{\right(x+1\left)\right(x+2\left)\right\}$, is

• A. 1
• B. 2
• C. 3
• D. 4

Answer 1

Answer: (C)

3

$f\left(x\right)={\log }_e\left\{\right(x+1\left)\right(x+2\left)\right\}$

For the function to be defined

$(x+1)(x+2)>0\Rightarrow -2>x>-1\Rightarrow x\in (-\infty ,-2)\cup (-1,\infty )$

$9x^2-18|x|+5=09{|x|}^2-18|x|+5=0\Rightarrow 9|{x|}^2-18|x|+5=0for(x>0)\Rightarrow 9|{x|}^2-3|x|-15|x|+5=0\Rightarrow 3|x|(3|x|-1)-5(3|x|-1)=0\Rightarrow (3|x|-1)(3|x|-5)=0\Rightarrow |x|=\frac{1}{3},\frac{5}{3}\Rightarrow x=\frac{1}{3},\frac{5}{3},-\frac{1}{3},-\frac{5}{3}$

Clearly $-\frac{5}{3}$ do not belong to the domain of the function $f\left(x\right)$

So there are $3$ solutions belonging to the domain of $f\left(x\right)$

Option $C$ is ccorrect.