[Algebra]
Problem: Compute the number of quadratic functions $f(x)=a x^{2}+b x+c$ with integer roots and integer coefficients whose graphs pass through the points $(0,0)$ and $(15,225)$.

Solution: Because the graph passes through $(0,0)$, conclude that $c=0$. Then

$$
f(15)=225 \Rightarrow a(15)^{2}+b(15)=225 a+15 b=225
$$

from which $b=15-15 a$. On the other hand, $f$ can be factored as $f(x)=a x(x+b / a)$, so if the roots are integers, $b / a$ must be an integer. Divide both sides of the equation $b=15-15 a$ by $a$ to obtain $b / a=15 / a-15$. Thus $15 / a$ must be an integer, and $a \in\{ \pm 1, \pm 3, \pm 5, \pm 15\}$. Because $b=15-15 a$ is linear, each of these values for $a$ yields a unique integer value for $b$, so there are 8 such ordered pairs. The values of $a, b$, and the nonnegative root are given in the table below.

| $a$ | $b$ | Root |
| ---: | ---: | ---: |
| 1 | 0 | 0 |
| 3 | -30 | 10 |
| 5 | -60 | 12 |
| 15 | -210 | 14 |
| -1 | 30 | 30 |
| -3 | 60 | 20 |
| -5 | 90 | 18 |
| -15 | 240 | 16 |

Final answer: 8