If $\mathrm{\alpha },\mathrm{\beta }$ are the roots of x²+bx+c = 0 and $(\mathrm{\alpha }+\mathrm{h}),(\mathrm{\beta }+\mathrm{h})$ are the roots of x²+qx+r = 0 then h =

Given $\alpha ,\beta$ are the roots of the equation $x^2+bx+c=0$

Use the relation between the roots and coefficients $\alpha +\beta =-b$

Given $\alpha +h,\beta +h$ are the roots of the eqution $x^2+qx+r=0$

use the relation bettween the roots and coefficients $\alpha +h+\beta +h=-q\Rightarrow \alpha +\beta +2h=-q$

It implies that 

    $\begin{array}{rcl}-b+2h& =& -q\\ 2h& =& b-q\\ h& =& \boxed{\frac{b-q}{2}}\end{array}$