Problem Of The Week #370 Jun 11th, 2019

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anemone

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Congratulations to the following members for their correct solution!

1. Olinguito
2. castor28
3. Cbarker1
4. Opalg

Solution from Opalg :
Any solution to the equation $x^3 - 3x = \sqrt{x+2}$ must satisfy $-2\leqslant x\leqslant 2$, because if $x<-2$ then the right side is not defined, and if $x>2$ then the left side is greater than the right.

Therefore all solutions must be of the form $x = 2\cos\theta$ for some $\theta$. Substitute that into the equation to get $$8\cos^3\theta - 6\cos\theta = \sqrt{2\cos\theta + 2},$$ $$2(4\cos^3\theta - 3\cos\theta) = \sqrt{2(\cos\theta + 1)},$$ $$2\cos(3\theta) = \sqrt{2(\cos\theta + 1)}.\qquad(*)$$ Now square both sides: $$4\cos^2(3\theta) = 2(\cos\theta+1),$$ $$2\cos^2(3\theta) - 1 = \cos\theta,$$ $$\cos(6\theta) = \cos\theta.$$ It follows that $6\theta = 2k\pi \pm\theta$ (for some integer $k$), so that either $\theta = \frac{2k\pi}5$ or $\theta = \frac{2k\pi}7$. But that includes several extraneous solutions arising from when the equation was squared. In fact, it follows from equation $(*)$ that we must have $\cos(3\theta)\geqslant0$. The only remaining values of $\theta$ to satisfy that are $\theta = 0$, $\frac{4\pi}5$ and $\frac{4\pi}7$. So the solutions of the original equation are $$x = 2\cos 0 = 2,$$ $$x = 2\cos\tfrac{4\pi}5 = -\tfrac12(\sqrt5 + 1),$$ $$x = 2\cos\tfrac{4\pi}7.$$

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