What is the minimum possible $ non $ integral value of a

Question

Let a A subscript(m) (m=1,2,3,....p) be the possible integral values of a for which the graphs of $ f(x)=ax^2+2bx+b $ and $g(x)=5x^2-3bx-a$ meets at some point for all real values of b.

1) What is the minimum possible $ non $ integral value of a

Answer 1

Assuming that $a \neq 5$, the solution of $f(x)=g(x)$ are given by $$x_{\pm}=\frac{-5b \pm\sqrt{25 b^2-4 (a-5) (a+b)}}{2 (a-5)}$$ The quantity under the radical must be greater or equal to $0$; after expanding this quantity, we then arrive to the condition $$-4 a^2-4 a (b-5)+5 b (5 b+4) \geq 0$$ The corresponding equation must have roots in $a$ (otherwise it will be negative because of the sign of the leading coefficient) and $a$ can only be between the roots. Solving this new quadratic equation leads to the roots $$a_{\pm}=\frac{1}{2} \left((5-b) \pm \sqrt{26 b^2+10 b+25}\right)$$ You could easily show that the quantity below the radical is always positive (no roots in $b$).

I am sure that you can take from here.