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Question
Find the quadratic polynomial, the sum of whose zeroes is `(5/2)` and their product is 1. Hence, find the zeros of the polynomial.
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Solution
Let α and β be the zeroes of the required polynomial f(x).
Then (α + β) = `5/2` and αβ = 1
∴ f(x) = x2 – (α + β)x + αβ
⇒ f(x) = `x^2 - 5/2 x + 1`
⇒ f(x) = 2x2 – 5x + 2
Hence, the required polynomial is f(x) = 2x2 – 5x + 2
∴ f(x) = 0 ⇒ 2x2 – 5x + 2 = 0
⇒ 2x2 – (4x + x) + 2 = 0
⇒ 2x2 – 4x – x + 2 = 0
⇒ 2x(x – 2) – 1(x – 2) = 0
⇒ (2x – 1) (x – 2) = 0
⇒ (2x – 1) = 0 or (x – 2) = 0
⇒ x = `1/2` or x = 2
So, the zeros of f(x) are `1/2` and 2.
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