The number of x[0,2] for which |√24x+18^2x - √24x+18^2x| = 1 is:

Question

The number of $x\in[0,2\pi]$ for which $\left|\sqrt{2\sin4x+18\cos^2x} - \sqrt{2\cos4x+18\sin^2x}\right| = 1$ is:

Accepted Answer

$$\sqrt{2\sin4x+18\cos^2x} - \sqrt{2\cos4x+18\sin^2x} = \pm1$$

$$\sqrt{2\sin4x+18\cos^2x} = \pm1+\sqrt{2\cos4x+18\sin^2x}$$

Squaring both sides gives a solvable equation in $x$; carrying this through over $[0,2\pi]$ yields 8 solutions.

Solutions · 1