The expression $y^2+10y+33$ can be written as a combination of a square of a binomial and an integer. Find the integer.
I want to write $y^2+10y+33$ as $(y+a)^2+b$, where $a$ and $b$ are integers. To do that, I need to complete the square for the quadratic term $y^2+10y$. To complete the square, I need to add and subtract the square of half the coefficient of $y$, which is $5^2=25$. So I have $y^2+10y+33=(y^2+10y+25)-25+33$. Now I can factor the perfect square binomial $y^2+10y+25=(y+5)^2$. So I have $y^2+10y+33=(y+5)^2-25+33=(y+5)^2+8$. Therefore, the expression $y^2+10y+33$ is written as a combination of a square of a binomial and an integer, where the integer is 8. 