<p>The number of real solutions of \(x^2 - |x| - 2 = 0\) is: [JEE Main 2023]</p>
Step-by-Step Solution
Key Concept: Let t = |x| \geq 0: t^2 - t - 2 = 0 \to (t-2)(t+1) = 0 \to t = 2 (t = -1 rejected). Then |x| = 2 \to x = \pm2. Two solutions.
Notice that the best first move is to reveal the hidden structure in the expression. A clever move here is to rewrite the problem in the form where the standard theorem or identity applies cleanly. Let $t=|x|\geq0$: $t^2-t-2=(t-2)(t+1)=0\Rightarrow t=2$ ($t=-1$ rejected since $t\geq0$). $|x|=2\Rightarrow x=\pm2$. Exactly 2 real solutions. Now, we invoke the power of that idea, simplify patiently, and then check that the final answer really fits the original problem.
Correct Answer: B