Question

If $\alpha ,\beta$ are the roots of $x^2+bx+c=0$ and $\alpha +h,\beta +h$ are the roots of $x^2+qx+r=0$, then $h$ is equal to

• A. $\frac{b+q}{2}$
• B. $b-q$
• C. $(b+q)$
• D. $\frac{(b-q)}{2}$

Answer 1

The correct option is D $\frac{(b-q)}{2}$

If the roots of the equation $a{x}^2+bx+c=0$ are $\alpha$ and $\beta$

Then $\alpha +\beta =\frac{-b}{a}$ ....(1)

and $\alpha \beta =-\frac{c}{a}$ .....(2)

Now, for quadratic equation $x^2+bx+c=0$ with roots $\alpha$ and

$\beta$ Using (1) and (2)

$\alpha +\beta =-b$

$\alpha \beta =c$

Similarly for $x^2+qx+r=0$ with roots $\alpha +h$ and $\beta +h$

$\alpha +h+\beta +h=-q$

$(\alpha +\beta )+2h=-q$

$-b+2h=-q$

$h=\frac{b-q}{2}$