Sum and product of the roots of a quadratic

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Question from Gautam, a parent:

Sir please help me find the solution to the problem given below
If a and b are the roots of x^2+px+1=0 and c and d are the roots of
x^2+qx+1=0
prove that
(a-c)(b-c)(a+d)(b+d) = q^2-p^2
Regards
Gautam

Gautam,

I looked for an easy way to see this but if there is one it eludes me. I just did it by brute force.

I do know that if a and b are roots of x² + px + 1 = 0 then a + b = -p and ab = 1. Similarly if c and d are rots of x² + qx + 1 = 0 then c + d = -q and cd = 1. Hence

q² - p² = (c + d)² - (a + b)²
= c² + 2cd + d² - a² - 2ab - b²
= c² + 2 + d² - a² - 2 - b²
= c² + d² - a² - b²

Now expand (a - c)(b - c)(a + d)(b + d) completely and use the sum of the roots and product of the roots above to simplify the expression and verify that its value is c² + d² - a² - b².

Harley

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