Question

How many roots do. the following equations possess?
3 |x| |2 - |x|| = 1.

Answer 1

$3\left|x\right||2-\left|x\right||=1$ (given)

$C1(0<x<2)$ $C2(x>2)$ $C3(-2<x<0)$

$3x(2-x)=1$ $-3x(2-x)=1$ $-3x(2+x)=1$

$\Rightarrow {3x}^2-6x+1=0$ $\Rightarrow {3x}^2-6x-1=0$ $\Rightarrow {3x}^2+6x+1=0$

$x=\frac{6\pm \sqrt{36-12}}{6}$ $x=\frac{6\pm 4\sqrt{3}}{6}$ $x=\frac{-6\pm 2\sqrt{6}}{6}$

$x=\frac{6\pm 2\sqrt{6}}{6}$ $x=\frac{6+4\sqrt{3}}{6}$

So, $5$ roots are there.