Question

If $2 \sin^3 x + \sin 2x \cos x + 4 \sin x - 4 = 0$ has exactly 3 solutions in the interval $\left[ 0, \frac{n \pi}{2} \right]$, $n \in \mathbb{N}$, then the roots of the equation $x^2 + nx + (n - 3) = 0$ belong to:

• A. $(0, \infty)$
• B. $(-\infty, 0)$
• C. $\left( -\frac{\sqrt{17}}{2}, \frac{\sqrt{17}}{2} \right)$
• D. $\mathbb{Z}$

Answer 1

Answer: (B)

$(-\infty, 0)$

Answer 2

Rewrite the given equation and analyze solution intervals to find values of $n$ for which the quadratic equation $x^2 + nx + (n - 3) = 0$ has roots in the desired interval.