If 23x+2x x+4 x-4=0 has exactly 3 solutions in the interval [0,n2], nN, then the roots of …

Question

If $2\sin3x+\sin2x\cos x+4\sin x-4=0$ has exactly 3 solutions in the interval $\left[0,\dfrac{n\pi}2\right]$, $n\in\mathbb{N}$, then the roots of the equation $x^2+nx+(n-3)=0$ belong to:

Accepted Answer

*(The source PDF supplies the stem, metadata, options, and answer key letter B, but the worked solution text was not present in the extraction, and attempting to re-derive the exact equation independently did not reproduce a clean value of $n$ consistent with verification --- so the derivation is not reproduced here to avoid presenting an unverified chain of reasoning as fact. The answer B, i.e. that the roots of $x^2+nx+(n-3)=0$ lie in $(-\infty,0)$, is taken directly from the source's answer key.)*

Solutions · 1