Factor fully $625-(y-2)^2$

Question

So far, I have used $(y-2)$ twice (multiplying both) because of the exponent being $2$. But, I need to factor and that's when I get confused. Please help!

Answer 1

Note that $625 - (y - 2)^2 = 25^2 - (y - 2)^2$. Let $z = y - 2$. Then $$625 - (y - 2)^2 = 25^2 - z^2 = (25 - z)(25 + z),$$ so $$625 - (y - 2)^2 = (25 - (y - 2))(25 + (y - 2)) = (27 - y)(23 + y).$$

Answer 2

Remember the difference of squares formula: $$a^2-b^2=(a+b)(a-b)$$ Our expression $625-(y-2)^2$ can be rewritten as: $$25^2-(y-2)^2$$ Set $a=25$ and $b=y-2$. Our expression can be factored as: $$(25+(y-2))(25-(y-2))$$ $$=(y+23)(27-y)$$ $$\displaystyle \boxed{625-(y-2)^2=(y+23)(27-y)}$$