Question

The number of distinct real roots of the equation \[ |x| \, |x + 2| - 5|x + 1| - 1 = 0 \] is _________.

Answer 1

Correct Answer: 3

Consider the different cases based on the value of \(x\).

Case 1: \(x \geq 0\)

\[ x^2 + 2x - 5x - 1 = 0 \implies x^2 - 3x - 6 = 0 \]

The roots are given by:

\[ x = \frac{3 \pm \sqrt{9 + 24}}{2} = \frac{3 \pm \sqrt{33}}{2} \]

Since \(x \geq 0\), one positive root exists.

Case 2: \(-1 \leq x < 0\)

\[ -x^2 - 2x - 5x - 1 = 0 \implies x^2 + 7x + 6 = 0 \]

The roots are:

\[ x = -1, \, x = -6 \]

Only \(x = -1\) is within the range.

Case 3: \(-2 \leq x < -1\)

\[ x^2 - 2x + 5x - 1 = 0 \implies x^2 - 3x - 4 = 0 \]

The roots are:

\[ x = \frac{-3 \pm \sqrt{9 + 16}}{2} = \frac{-3 \pm \sqrt{25}}{2} \]

No root lies in the range.

Case 4: \(x < -2\)

\[ x^2 + 7x + 4 = 0 \]

The roots are:

\[ x = \frac{-7 \pm \sqrt{49 - 16}}{2} = \frac{-7 \pm \sqrt{33}}{2} \]

One root lies in the range.

Total number of distinct real roots: 3