1. We need to solve $x^2-|x|-12=0$ and count the number of real solutions.

2. Since the equation contains $|x|$, split into cases.

Case 1: $x \ge 0$

Then $|x|=x$. So the equation becomes $x^2-x-12=0$ Factorizing, $x^2-x-12=(x-4)(x+3)=0$ So the roots are $x=4,\,-3$ But in this case we require $x\ge 0$, so only $x=4$ is valid.

Case 2: $x<0$

Then $|x|=-x$. So the equation becomes $x^2-(-x)-12=0 \implies x^2+x-12=0$ Factorizing, $x^2+x-12=(x+4)(x-3)=0$ So the roots are $x=-4,\,3$ But in this case we require $x<0$, so only $x=-4$ is valid.

3. Therefore, the real solutions are $x=4 \quad \text{and} \quad x=-4$ Hence, the number of real solutions is $2$

4. Checking options:

• A: $2$ ✅
• B: $3$ ❌
• C: $1$ ❌
• D: $4$ ❌

So the correct option is A.