If alpha, beta be roots of equation x^2 - px + q = 0, then form an equation whose roots are q/(p - alpha) and q/(p - beta)

Okay. So lets begin with remarking this as a good question.

We are given the equation x^2 - px + q = 0 and it is said that its roots are α and β. We have to form an equation whose roots are q/(p - α) and q/(p - β). This equation which has to be found has two roots, so obviously it is also a quadratic equation.

The approach to these types of questions is very basic. First of all, we'll apply the Sum and Product of roots formulae on the given equation.

Sum of roots = -(-p)/1

=> α + β = p               (1)

Product of roots = q/1

=> αβ = q                   (2)

We know that any Quadratic Equation can be written in the form x^2 - (sum of its roots)x + product of its roots = 0. So, to find the equation whose roots are q/(p - α) and q/(p - β), we'll have to find the sum and product of these roots.

Sum of roots

= q/(p - α) + q/(p - β)

= q [1/(p - α) + 1/(p - β)]

= q [ (p - β + p - α) / (p - α)(p - β)]

= q [(2p - (α + β)) / (p^2 - pα - pβ + αβ)]

= q [(2p - (α + β)) / (p^2 - p(α + β) + αβ)]

Now, using (1) and (2),

= q [(2p - p) / (p^2 - p(p) + q)]

= q ( p / p^2 - p^2 + q )

= q (p/q)

= p

Product of roots

= q/(p - α) * q/(p - β)

= q^2 / (p - α)(p - β)

= q^2 / (p^2 - pα - pβ + αβ)

= q^2 / (p^2 - p(α + β) + αβ)

Again, using (1) and (2)

= q^2 / (p^2 - p^2 + q)

= q^2 /q

= q

We've found out that Sum of roots of the desired equation is pq and the Product of roots for the same is q^2. Thus, we can say that the equation is

x^2 - (p)x + q = 0

=> x^2 - px + q = 0

Hence, the desired equation is x^2 - px + q = 0. Interestingly, this is the original equation itself! And they said that maths in boring?

  Additional Explanation 

You could probably wonder if it had been possible to 'guess' the answer at an earlier stage. After all, it was said that x^2 + px - q = 0 had roots α and β, and then when you found an equation whose roots were q/(p - α) and q/(p - α), the equation came out to be the same old x^2 + px - q = 0 ! Now, since a quadratic equation has only two roots, all we can say now is that "α and β equals q/(p - α) and q/(p - β)". NOTE THAT WHEN WE SAY THAT, WE MEAN THAT EITHER α = q/(p - β) and β = q/(p - α), OR α = q/(p - α) and β = q/(p - β). Actually, it was the same pair of roots represented in a 'messed up' way.

Now, the question. Could you have guessed it? Before I answer it, have a look at how you can find out that they 'messed up' roots given are the same old α and β.

The equation x^2 - px + q = 0 has roots α and β. They will satisfy the equation. So, we can say that α^2 - pα + q = 0 and β^2 - pβ + q = 0. Let us try something with the first result.

α^2 - pα + q = 0

=> q = pα - α^2

=> q = α(p - α)

=> q/(p - α) = α

If we follow the same route with the second result, it'll lead us to q/(p - β) = β.

Here it is. They were the same α and β written as q/(p - α) and q/(p - β). Could you've guessed it this way before? Why would anyone experiment with a question when he/she already knows the right direction to the solution, unless they reach to an interesting result in the end and then find the question worth some time. As for me, I did began solving it the regular way like any other question and later, when I reached the answer, I gave it a thought. So, I think that any average human who studies maths could've guessed it only if he/she had been through a similar situation before.