The number of distinct real roots of the equation \[ |x| \, |x + 2| - 5|x + 1| - 1 = 0 \] is _________.
Correct Answer: 3
Solution and Explanation
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
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