Distributing exponents to variables

Question

I am studying parabolas, but the way exponents are distributed is confusing me to oblivion.

$(y_2 – 2)^2$

$y_2^2 – 4y_2^2 + 4$

I do not understand how can the first expression be simplified to the next. Could someone please provide an explanation?

Answer 1

In general, $$(a+b)^2 = (a+b)(a+b)= aa+ab+ba+bb = a^2+2ab+b^2.$$

The result in your question will follow if you let $a=y_2$ and $b=-2$.

$$(y_2+(-2))^2=y_2^2+2(-2)y_2+(-2)^2=y_2^2-4y_2+4.$$

Answer 2

$(a-b)^{2}= a^{2}-2ab+b^2$. Here $a = y_{2}$ and $b=2$.