Question

The product of the roots of the equation $9x^2-18\left|x\right|+5=0$ is

• A. $\frac{25}{81}$
• B. $\frac{5}{9}$
• C. $\frac{5}{27}$
• D. $\frac{25}{9}$

Answer 1

The correct option is A

$\frac{25}{81}$

Explanation for correct option:

Find the product of the roots of the given equation

Given, $9x^2-18\left|x\right|+5=0$

$$\begin{gathered} 9x^2-18\left|x\right|+5=0 \\ \Rightarrow 9{\left|x\right|}^2-18\left|x\right|+5=0 \\ \Rightarrow 9{\left|x\right|}^2-15\left|x\right|-3\left|x\right|+5=0 \\ \Rightarrow 3\left|x\right|\left(3\left|x\right|-5\right)-1(3\left|x\right|-5)=0 \\ \Rightarrow (3\left|x\right|-1)(3\left|x\right|-5)=0 \\ \Rightarrow \left|x\right|=\frac{1}{3},\frac{5}{3} \\ \Rightarrow x=\pm \frac{1}{3},\pm \frac{5}{3} \end{gathered}$$

Zeroes of $9x^2-18\left|x\right|+5=0$are $x=\pm \frac{1}{3},\pm \frac{5}{3}$.

Compute the product $P$ of the roots as follows:

$$\begin{gathered} P=\frac{1}{3}\times \left(-\frac{1}{3}\right)\times \frac{5}{3}\times \left(-\frac{5}{3}\right) \\ \Rightarrow P=\frac{25}{81} \end{gathered}$$

Therefore, the product of the roots of the given equation is $\frac{25}{81}$.

Hence, option (A) is the correct answer.