The graphs of $y = x^2 - 7x + 7$ and the line $y = -3$ intersect at two points. What is the sum of the $x$-coordinates of these two points?
I want to find the values of $x$ that satisfy both equations simultaneously, so I can set them equal to each other and solve for $x$. That is, I have $x^2 - 7x + 7 = -3$. I can simplify this by adding $3$ to both sides and get $x^2 - 7x + 10 = 0$. Now I have a quadratic equation that I can try to factor. I look for two numbers that multiply to $10$ and add to $-7$. I see that $-5$ and $-2$ work, so I can write $x^2 - 7x + 10 = (x - 5)(x - 2)$. This means that either $x - 5 = 0$ or $x - 2 = 0$, so $x = 5$ or $x = 2$. These are the two $x$-coordinates of the points of intersection. To find their sum, I just add them together: $5 + 2 = 7$. 