Question

Roots(s) of the equation $9x^2-18|x|+5=0$ belonging to the domain of definition of the $f\left(x\right)=\log (x^2-x-2)$ is (are)

• A. $-\frac{5}{3},-\frac{1}{3}$
• B. $\frac{5}{3},\frac{1}{3}$
• C. $-\frac{5}{3}$
• D. $-\frac{1}{3}$

Answer 1

Answer: (C)

$-\frac{5}{3}$

$9x^2-18\left|x\right|+5=09{\left|x\right|}^2-15\left|x\right|-3\left|x\right|+5=03\left|x\right|[3\left|x\right|-5]-1[3\left|x\right|-5]=0(3\left|x\right|-5)(3\left|x\right|-1)=0$

So, $\left|x\right|=\frac{5}{3},\frac{1}{3}$

$x=\pm \frac{5}{3},\pm \frac{1}{3}$

Now, for $f\left(x\right)$ to be real,

$x^2-x-2>0(x-2)(x+1)>0xϵ(-\infty .-1)\cup (2,\infty )$

So, $x=\frac{-5}{3}$