Animated Solution for Mathematics - Quadratic Equations: The number of real solutions of the equation, $x^2-|x|-12=0$ is:

The equation provided is $x^2-|x|-12=0$. While it resembles a standard quadratic equation, the presence of the absolute value term $|x|$ introduces a specific constraint that we must address carefully.

The first step is to recognize the relationship between $x^2$ and $|x|$. For any real number $x$, the square of the number is identical to the square of its absolute value:

By substituting this identity into our original equation, we transform it into a quadratic form involving only the absolute value:

To simplify the algebra, let us introduce a substitution variable. Let $t=|x|$.

Our equation now becomes a standard quadratic:

However, we must respect the gatekeeper constraint. Since $t=|x|$ and the absolute value of any real number is non-negative, we must enforce the condition:

We now factor the quadratic equation $t^2-t-12=0$. We look for two numbers that multiply to $-12$ and add to $-1$. These numbers are $-4$ and $3$.

The factored form is:

This yields two potential roots for $t$:

We must now check these roots against our constraint $t\ge 0$. The value $t=-3$ is rejected because an absolute value cannot be negative.

This leaves us with the single valid solution:

Substituting back for $x$, we have $|x|=4$. This implies that $x$ can take two possible values: