If $\alpha, \beta$ are roots of the equation $x^2 + px – q = 0$. $\gamma , \delta$ are roots of equation $x^2 + px -r$, then find the value of $(\alpha – \gamma )(\alpha – \delta)$.
Answer – $q-r$
My try –
$\alpha + \beta= -p$ and $\alpha \beta = -q$
similarly,
$\gamma + \delta = -p$ and $\gamma \delta = -r$
Then to we've to find:
$(\alpha – \gamma)(\alpha – \delta) = \alpha^2 – \alpha \delta – \alpha \gamma + \gamma \delta $ out of which only $\gamma \delta$ is known, then how to find the rest?
Also, when noticed carefully about the question, we find that question is $(\alpha – \gamma)(\alpha – \delta)$ which doesn't have $\beta$ in its product, which makes the question more confusing.
Thanks in Advance 🙂
Best Answer
We have $\alpha + \beta = \gamma + \delta \implies \beta = \gamma + \delta - \alpha$
Now, $(\alpha - \gamma)(\alpha - \delta) = \alpha^2 - \alpha \delta - \alpha \gamma + \gamma \delta = \alpha(\alpha - \delta - \gamma) + \gamma\delta = \alpha(-\beta) +\gamma\delta = \gamma\delta - \alpha\beta $
$= -r + q$