Animated Solution for Mathematics - Quadratic Equations: The product of the roots of the equation $9x^2-18|x|+5=0$, is:

When you look at the equation $9x^2-18|x|+5=0$, you see a quadratic with a mask on. That mask is the absolute value.

Because $|x|$ is the same as $|-x|$, the function is perfectly symmetric across the $y$-axis. This tells us that if a positive number $x$ is a solution, its negative counterpart $-x$ must also be a solution.

To strip away the mask, we use the identity $x^2=|x{|}^2$. Whether you square $3$ or $-3$, you get $9$, and the same holds for their absolute values.

This identity allows us to rewrite the equation entirely in terms of $|x|$:

To simplify, let $t=|x|$. Since $t$ represents an absolute value, we must enforce the constraint $t\ge 0$.

Substituting $t$ into our equation yields:

We factor this quadratic by finding two numbers that multiply to $45$ and add to $-18$, which are $-15$ and $-3$:

We find two potential values for $t$: $t=\frac{1}{3}$ and $t=\frac{5}{3}$. Since both are positive, both are valid.

We now reverse the substitution $|x|=t$:

1. If $|x|=\frac{1}{3}$, then $x=\pm \frac{1}{3}$. 2. If $|x|=\frac{5}{3}$, then $x=\pm \frac{5}{3}$.

The four roots of the equation are $\frac{1}{3},-\frac{1}{3},\frac{5}{3},$ and $-\frac{5}{3}$.

The product of these roots is calculated as follows:

Multiplying these values together:

The final product of the roots is $\frac{25}{81}$.