Where is the $-2$ coming from?

Question

If $y=x +\frac 1x$, how is $x^2 + \frac1{x^2} = y^2 - 2$. Shouldn't the answer be just $y^2$, where is the $-2$ coming from.

Answer 1

It is $$y^2=x^2+2\times x\times\frac{1}{x}+\frac{1}{x^2}$$

Answer 2

$$\left(x+\frac{1}{x}\right)^2 = x^2+2\frac{x}{x}+\frac{1}{x^2} =x^2+2+\frac{1}{x^2}$$

Answer 3

Because $$y^2=\left(x+\frac{1}{x}\right)^2=x^2+\frac{1}{x^2}+2.$$ We used $(a+b)^2=a^2+b^2+2ab.$

Answer 4

Note that $$(a+b)^2=a^2+b^2+2ab$$

With the above formula in mind try to expand $$y^2=(x+\frac {1}{x})^2 =x^2+2+\frac {1}{x^2}$$ and you find your $-2$