If the solution for $(7-k)(8-k)x^2 - (112-15k)x + 56 = 0$ are all integer solutions, what could the value for $k$ be? (There are 4 values)
My first thought was to put in the quadraic formula and see what the solution will look like.
So put in: $$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$$ Where we'll get $$x = \frac{112-15k\pm \sqrt{337k^2-5020k+18816}}{2(7-k)(8-k)}$$ after simplification.
However, I found that $337k^2-5020k+18816$ cannot be factorized. What should I do?
You didn't solve the quadratic equation in $x$ correctly. I obtained $$ x=\frac{7}{7-k},\quad x=\frac{8}{8-k}. $$ Indeed, $b^2-4ac=k^2$.