If √3^2x = (√3-1) x+1, the number of solutions of the given equation when x[0,2] is ______…

Question

If $\sqrt3\cos^2x = (\sqrt3-1)\cos x+1$, the number of solutions of the given equation when $x\in\left[0,\dfrac\pi2\right]$ is \_\_\_\_\_\_\_\_\_\_\_

Answer: 1

Accepted Answer

Let $t=\cos x$: $\sqrt3t^2-(\sqrt3-1)t-1=0 \implies \sqrt3t^2-\sqrt3t+t-1=0 \implies \sqrt3t(t-1)+(t-1)=0 \implies (t-1)(\sqrt3t+1)=0$

$$t=1 \quad\text{or}\quad t=-\frac1{\sqrt3}$$

Since $x\in\left[0,\dfrac\pi2\right]$, $\cos x\ge0$, so $t=-\tfrac1{\sqrt3}$ is rejected. Only $\cos x=1$, i.e. $x=0$, is valid.

$$\text{Number of solutions} = 1$$

Solutions · 1