Question

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

Answer 1

`abs(x^2-2)=abs(x)` has `color(green)(4)` solutions

Explanation:

`abs(x^2-x)=abs(x)`
`rArr`
#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:
`color(white)("XXX")x in {-2, -1, +1, +2}`

Answer 2

Graph reveals solutions `x = +-1 and x = +-2`..

Explanation:

The graphs `y = |x| and y =|x^2-2|` intersect at `x = +-1 and x = +-2`.

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

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

piecewise definitions, sans `|...|` 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]}