How do you solve a system of equations in the form $x+y=3, xy=4$?

Question

I am trying to factor this polynomial: $x^2+3x+4$. and couldn't think of any integer solution (I don't think there is a real solution either) so I created this system of equations. $x+y=3$ and $xy=4$. The only problem is, I have no idea how to solve a system of this form. Please use terms that an Algebra 1 student would understand.

Answer 1

Let $s$ be the sum and $p$ the product.

Then write $x=\dfrac s2+z$ and $y=\dfrac s2-z$, so that $x+y=s$, and from $xy=p=\dfrac{s^2}4-z^2$ you can draw $z$.