I've gone through my workings twice and re-did, but somehow I keep getting an irregularity.

I did it like this.

Since the roots are $\displaystyle \alpha$ and $\displaystyle \alpha + 5$, I substituted them into the equation.

$\displaystyle \alpha^2 - (q - 5)\alpha = (\alpha + 5)^2 - (q - 5)(\alpha + 5)$

$\displaystyle \alpha^2 - (q - 5)\alpha = \alpha^2 + 10\alpha + 25 - (q\alpha - 5\alpha + 5q - 25)$

$\displaystyle \alpha^2 - (q - 5)\alpha = \alpha^2 | 10\alpha + 25 - q\alpha + b\alpha - 5q + 25)$

[tex]\alpha^2 - (q - 5)\alpha = \alpha^2 + (15 - q)\alpha + (50 - 5q)[tex]

Therefore $\displaystyle -(q - 5)\alpha = (15 - q)\alpha$, $\displaystyle 50 - 5q = 0$

$\displaystyle 5 - q = 15 - q$, $\displaystyle q = 10$

But 5-q = 15-q doesn't even make any sense.