How many solutions does $\sin^2x+3\sin x\cos x+2\cos^2 x=0$ have?

How many solutions does $\sin^2x+3\sin x\cos x+2\cos^2 x=0$ have?

Question

In the range $0\leq x<2\pi$, the equation $\sin^2x+3\sin x\cos x+2\cos^2 x=0$ has

1. $1$ solution,

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1. $2$ solutions,

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1. $3$ solutions,

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1. $4$ solutions.

Solution

If $\cos x = 0, \sin x \neq 0$, so the equation does not hold. Thus we are free to divide by $\cos^2x$, since it must be non-zero, giving $\tan^2x + 3\tan x + 2=0,$

which factorises into $(\tan x +1)(\tan x +2 ) = 0.$

So our solutions will be those for $\tan x = -1, \tan x = -2.$

Since $y=\tan x$ has a period of $\pi$, there will therefore be four solutions in total, which means the answer is (d).