Find all solutions when two functions are the same (same result)

Question

Can anybody help me find solutions for $x$ when: $$y_1 = y_2$$ and equations are (for example): $$y_1 = \frac{(x^2+x)}{2} \;; \;y_2 = \frac{(x^2+19x-12)}{2}$$

By solving it "brute force" I can tell some of the solutions are:

1. $$y_1(5)=y_2(2)=15$$
2. $$y_1(15)=y_2(9)=120$$
3. $$?\ \ \ \ ? \ \ \ ?$$

Thank you!

Answer 1

HINT:

For integers $a,b$, we wish to solve $$\frac{a^2+a}2=\frac{b^2+19b-12}2\implies\left(a+\frac12\right)^2-\frac14=\left(b+\frac{19}2\right)^2-\frac{361}4-12$$ so $$\left(b+\frac{19}2\right)^2-\left(a+\frac12\right)^2=102\implies (a+b+10)(b-a+9)=2\cdot3\cdot17$$ from which only eight possible combinations exist.