Given equation

$9x^2 - 18|x| + 5 = 0$

We need the product of all roots of this equation.

Substitute

Let $y = |x|$ Then since $x^2 = |x|^2 = y^2$, the equation becomes $9y^2 - 18y + 5 = 0$

Solve the quadratic in $y$

$9y^2 - 18y + 5 = 0$

Using factorization or quadratic formula: $9y^2 - 18y + 5 = (3y-1)(3y-5)=0$

So, $y = \frac{1}{3}, \quad y = \frac{5}{3}$

Convert back to $x$

Since $y = |x|$,

• If $|x| = \frac{1}{3}$, then $x = \pm \frac{1}{3}$
• If $|x| = \frac{5}{3}$, then $x = \pm \frac{5}{3}$

Therefore, the roots are $\frac{1}{3},\ -\frac{1}{3},\ \frac{5}{3},\ -\frac{5}{3}$

Find their product

$\left(\frac{1}{3}\right)\left(-\frac{1}{3}\right)\left(\frac{5}{3}\right)\left(-\frac{5}{3}\right)$

First pair: $\left(\frac{1}{3}\right)\left(-\frac{1}{3}\right) = -\frac{1}{9}$

Second pair: $\left(\frac{5}{3}\right)\left(-\frac{5}{3}\right) = -\frac{25}{9}$

Thus, $\left(-\frac{1}{9}\right)\left(-\frac{25}{9}\right)=\frac{25}{81}$

Check options

$\frac{25}{81}$ corresponds to Option C.

Comparison with stored answer

Stored correct answer: C

Our derived answer: C

Hence, they agree.