Question

If α and β are the zeros o the quadratic polynomial f(x)= x2-px+q, then find the values of : α^2 +β^2 (ii) 1/α+1/β

Answer 1

α^2 +β^2
=(alpha+bita)²-2(alpha. bita)
=p²-2q

..&
2
(alpha+bita)/(alpha.bita)
=p/q

Answer 2

Solution:

We have,

f(x) = x2 + px + q

Sum of the zeroes = α + β = -p

Product of the zeroes = αβ = q

From the question,

Sum of the zeroes of new polynomial = (α + β)2 + (α – β)2

= (α + β)2 + α2 + β2 – 2αβ

= (α + β)2 + (α + β)2 – 2αβ – 2αβ

= (- p)2 + (- p)2 – 2 × q – 2 × q

= p2 + p2 – 4q

= p2 – 4q

Product of the zeroes of new polynomial = (α + β)2 (α – β)2

= (- p)2((- p)2 - 4q)

= p2 (p2–4q)

So, the quadratic polynomial is,

x2 – (sum of the zeroes)x + (product of the zeroes)

= x2 – (2p2 – 4q)x + p2(p2 – 4q)

Hence, the required quadratic polynomial is f(x) = k(x2 – (2p2 –4q) x + p2(p2 - 4q)).