Sum and Product Of Roots

Question

If α and β are the roots of the equation x² + px -q =0 and γ and δ are roots of x² +px+r =0 then the value of (α-γ)(α-δ) is:- (Choose the correct option)

1. p+r
2. p-r
3. q-r
4. q+r

Answer 1

As $\alpha$ is one of its roots (first equation) and we know the value of an expression at its roots is equal to zero so --> $\alpha^2+p\alpha-q =0$ or $\alpha^2+p\alpha=q$ //result 1

Now in second equation sum of roots $=-p=\gamma+\delta$//result 2 and product of roots $\gamma\delta=r$//result 3

Now comes the required equation $(\alpha-\gamma)(\alpha-\delta)$

Opening the brackets we get $\alpha^2-\alpha(\gamma+\delta)+\gamma\delta$.

Put result 2 and 3 here so we get $\alpha^2+p\alpha+r$

Now use result 1 and put $\alpha^2+p\alpha=q$ to get your answer $q+r$.