Let . Find all real numbers such that .

Let . Find all real numbers  such that . 

So...we want to find this....

x² - 2x  = (x² - 2x)² - 2(x² - 2x)         expand

x² - 2x = x⁴ - 4x³ + 4x² - 2x² + 4x     simplify

x² - 2x = x⁴ - 4x³ + 2x²  + 4x            subtract x² - 2x from both sides

0 = x⁴ - 4x³ + x²  + 6x                      rearrange and factor

x (x³ - 4x² + x + 6) = 0

One solution is x = 0

The others are shown in this graph......https://www.desmos.com/calculator/pfvqpci5oa

And the other solutions are x = -1   , x = 2 and x = 3

So, the solurions in order are

x = -1, 0, 2 and 3

wait, is that negative POINT 838? or is it just -838? Sorry!

My teacher said that this answer was wrong.

Sorry...I made a slight mis-type when after I factored the polynomial.......see the corrected answer in my response.....it is correct!!

The answers are -1, 0,2, 3 right?