Question

For the equation $|x^2|+|x|-6=0$, the roots are

• A. One and only one real number
• B. Real with sum one
• C. Real with sum zero
• D. Real with product zero

Answer 1

The correct option is C

Real with sum zero

When x < 0, |x| = -x

∴ Equation is $x^2-x-6=0$ ⇒ x = -2, 3

∵ x < 0, ∴ x = -2 is the solution.

When x ≥ 0, |x| = x

∴ Equation is $x^2+x-6=0$ ⇒ x = 2, -3

∵ x ≥ 0, ∴ x = 2 is the solution.

Hence x = 2, -2 are the solutions and their sum is zero.

Aliter : $|x^2|+|x|-6=0$

⇒ (|x| + 3)(|x| - 2) = 0

⇒ |x| = -3, which is not possible and |x| = 2

⇒ x = ± 2.