Question

Write the number of real roots of the equation x² + 3 |x| + 2 = 0.

Answer 1

Key concept |x| = -x for x < 0

|x| = +x for x > 0

For case I: x < 0

⇒ x² - 3x + 2 = 0

⇒ x² - (2 + 1)x + 2 = 0

⇒ x² - 2x - x + 2 = 0

⇒ x(x - 2) -1 (x - 2) = 0

⇒ (x - 1) (x - 2) = 0

⇒ x = 1, 2

But for |x| = -x i.e x < 0

Here value of x = 1, 2 is > 0

So, condition does not satisfy.

For x > 0 i.e |x| = +x

x² + 3x + 2 = 0

⇒ x² + (2 + 1)x + 2 = 0

⇒ x² + 2x + x + 2 = 0

⇒ x (x + 2) +1 (x + 2) = 0

⇒ (x + 1) (x + 2) = 0

⇒ [x = -1, -2]

Again x < 0 here and we know that |x| = +x for x > 0

So, condition does not satisfy.

Therefore, No real roots exist.