$(-x+2)^2=(x-2)^2$ this is true but why?

Question

I came across this identity in an old textbook:

$$(-x+2)^2=(x-2)^2$$

My only problem is that it does not feel natural. It is hard for me to visualize why this is true. A simple explanation on how to think about this problem would be really nice. Thanks in regard.

Answer 1

Opposite numbers have the same square, for example: $$4^2 = (-4)^2 = 16$$ This holds for any real number $a$, symbolically: $$a^2 = (-a)^2$$ So it also holds for $x-2$; take $a=x-2$ in the formula above and you have: $$(x-2)^2 = \bigl( -(x-2) \bigr)^2 = (2-x)^2$$

Answer 2

$$\begin{align}(x-2)=(-1)(-x+2)\\(x-2)^2=(-1)^2(-x+2)^2\\(x-2)^2=(-x+2)^2\end{align}$$

Answer 3

When you square something, there is no matter with what sign you are taking it, for example: $n^2=(-n)^2$. It applies to every even number in the power $n^4=(-n)^4$ and so on... In our example, $n = x - 2$ and $-n = -2 + x$.