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# Solution - If 𝛼 and 𝛽 Are the Zeros of the Quadratic Polynomial F(X) = X2 − 3x − 2, Find a Quadratic Polynomial Whose Zeroes Are 1/(2alpha+Beta)+1/(2beta+Alpha) - CBSE Class 10 - Mathematics

ConceptRelationship Between Zeroes and Coefficients of a Polynomial

#### Question

If α and β are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2, find a quadratic polynomial whose zeroes are 1/(2alpha+beta)+1/(2beta+alpha)

#### Solution

Since α and β are the zeros of the quadratic polynomial f(x) = x2 − 3x − 2

The roots are α and β

alpha+beta="-coefficient of x"/("coefficient of "x^2)

alpha+beta=-((-3)/1)

α + β = -(-3)

α + β = 3

alphabeta="constant term"/("coefficient of "x^2)

alphabeta=(-2)/1

αβ = -2

Let S and P denote respectively the sum and the product of zero of the required polynomial . Then,

S=1/(2alpha+beta)+1/(2beta+alpha)

Taking least common factor then we have ,

S=1/(2alpha+beta)xx(2beta+alpha)/(2beta+alpha)+1/(2beta+alpha)xx(2alpha+beta)/(2alpha+beta)

S=(2beta+alpha)/((2alpha+beta)(2beta+alpha))+(2alpha+beta)/((2beta+alpha)(2alpha+beta))

S=(2beta+alpha+2alpha+beta)/((2alpha+beta)(2beta+alpha))

S=(3beta+3alpha)/(4alphabeta+2beta^2+2alpha^2+betaalpha)

S=(3(beta+alpha))/(5alphabeta+2(alpha^2+beta^2))

S=(3(beta+alpha))/(5alphabeta+2[(alpha+beta)^2-2alphabeta])

By substituting α + β = 3 and αβ = -2 we get,

S=(3(3))/(5(-2)+2[(3)^2-2xx-2])

S=9/(-10+2(13))

S=9/(-10+26)

S=9/16

P=1/(2alpha+beta)xx1/(2beta+alpha)

P=1/((2alpha+beta)(2beta+alpha))

P=1/(4alphabeta+2beta^2+2alpha^2+betaalpha)

P=1/(5alphabeta+2(alpha^2+beta^2))

P=1/(5alphabeta+2[(alpha+beta)^2-2alphabeta])

By substituting α + β = 3 and αβ = -2 we get,

P=1/(5(-2)+2[(3)^2-2xx-2])

P=1/(10+2[9+4])

P=1/(10+2(13))

P=1/(-10+26)

P=1/16

Hence ,the required polynomial f(x) is given by

f(x) = k(x^2 - Sx + P)

f(x) = k(x^2-9/16x+1/16)

Hence, the required equation is f(x) = k(x^2-9/16x+1/16) Where k is any non zero real number.

Is there an error in this question or solution?

#### Reference Material

Solution for question: If 𝛼 and 𝛽 Are the Zeros of the Quadratic Polynomial F(X) = X2 − 3x − 2, Find a Quadratic Polynomial Whose Zeroes Are 1/(2alpha+Beta)+1/(2beta+Alpha) concept: Relationship Between Zeroes and Coefficients of a Polynomial. For the course CBSE
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