What is the number of solutions of the equation abs(x^2-2)=absx?

Dec 27, 2016

$\left\mid {x}^{2} - 2 \right\mid = \left\mid x \right\mid$ has $\textcolor{g r e e n}{4}$ solutions

Explanation:

$\left\mid {x}^{2} - x \right\mid = \left\mid x \right\mid$
$\Rightarrow$
color(white)("XXX"){:("Either",," or ",), (,x^2-2=x,,x^2-2=-x), (,x^2+x-2=0,,x^2+x-2=0), (,(x+2)(x-1)=0,,(x-2)(x+1)=0), (,x=-2 or +1,,x=+2 or -1) :}

So there are 4 possible solutions:
$\textcolor{w h i t e}{\text{XXX}} x \in \left\{- 2 , - 1 , + 1 , + 2\right\}$

Dec 27, 2016

Graph reveals solutions $x = \pm 1 \mathmr{and} x = \pm 2$..

Explanation:

The graphs $y = | x | \mathmr{and} y = | {x}^{2} - 2 |$ intersect at $x = \pm 1 \mathmr{and} x = \pm 2$.

So, these are the solutions of (x-2|=|x|.

Of course, algebraically, these solutions can be obtained, using

piecewise definitions, sans $| \ldots |$ symbol.

Note of caution: In general, graphical solutions are approximations

only.

graph{(y-|x|)(y-|x^2-2|)=0x^2 [-5, 5, -2.5, 2.5]}