# If & are two zeros of the quadratic polynomial f(x)=-3x-2,find a quadratic polynomial whose zeros are 1/2 alpha+beta & 1/2 beta+alpha

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by Aparna8

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by Aparna8

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general equation of a quadratic function = x^2 - (α+β)x+αβ

then, in given Eqn. α+β = 3 ---------(1) & αβ = -2 --------------(2)

we know that, (α+β)^2-(α-β)^2 = 2αβ

(3)^2 - (α-β)^2 = 2·(-2)

⇒ (α-β)^2 = 13

⇒α-β = √13 -------------(3)

EqN(1) + EqN(3)

α = (3+√13)/2

β = -2/α = -4/(3+√13)

Now.

New Eqn will be ,whose zero's are α/2+β & α+β/2

x^2- (α/2+β)+αβ/2

⇒ x^2-(α+2β)/2+αβ/2

⇒ x^2-(α+β+β)/2+αβ/2

⇒ x^2 - {3-4/(3+√13)}/2 -1 -----------(α+β)= 3 & β = -4/(3+√13).,, ,αβ= -2

if u solve this eqn then ,u find a equation