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Question

If α and β are the zeros of the quadratic polynomial f(x)=x25x+6, then evaluate:

α2β2+β2α2
[4 MARKS]

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Solution

Concept : 1 Mark
Application : 1 Mark
Calculation : 2 Marks

Since α and β are the zeros of the quadratic polynomial f(x)=ax2+bx+c

α+β=ba=5 and αβ=ca=6
We have α+β=5
Hence (α+β)2=25
α2+β2+2αβ=25
α2+β2=252αβ
α2+β2=252(6) ( αβ=6) = 13

We have,

α2β2+β2α2=α4+β4α2β2
α2β2+β2α2=α2β2+β2α2=(α2+β2)22α2β2α2β2
=132(2×(62))62
=9736.
α2β2+β2α2=9736.

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