The value of parameter a for which $\frac{ax^2+3x−4}{a+3x−4x^2}$ takes all real values for $x\in \mathbb R$ are:

Question

I solve it and get the answer $a\in [1,7]$.but my teacher told me to take the verification of the boundary values of a.because at the boundary values, $ax^2+3x-4$ and $3x-4x^2+a$ have common roots.so,its obvious that value of a will not count in the answer.But my question is,is it necessary to have common roots between $ax^2+3x-4$ and $3x-4x^2+a$ at only boundary values.if yes then why??