A student tries to use the quadratic formula on $5x-7=0$, taking $a=0,b=5$ and $c=-7$.
Why does this fail?

The quadratic formula is stated for $a{x}^2+bx+c=0$ with a ≠ 0 precisely because $2a$ sits in the denominator. With $a=0$ that denominator is 0 and the expression is meaningless. $5x-7=0$ is a linear equation with the single solution $x=\frac{7}{5}$.