The number of real solutions of the equation, x² - |x| - 12 = 0 is :

Solution

Maths sir · 4 steps

Put t = |x|: since x² = |x|², this is an ordinary quadratic in t, and t cannot be negative.

1. Substitute. Because x² = |x|², let t = |x| with t ≥ 0. The equation becomes t² - t - 12 = 0.

   t² - t - 12 = (t - 4)(t + 3) = 0

2. Keep valid roots. So t = 4 or t = -3. Since t = |x| can never be negative, reject t = -3 and keep only |x| = 4.
3. Back to x. |x| = 4 gives x = 4 or x = -4. Check: 16 - 4 - 12 = 0 for both.
4. So. There are exactly 2 real solutions, x = ± 4, which is option (a).

The idea to keep. When x² and |x| appear together, put t = |x|. Solve the quadratic in t, throw away negative roots, and each positive root gives two values of x.

Answer from the NTA final answer key. Solution by PrepWiser.