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Question
If a and 3 are the zeros of the quadratic polynomial f(x) = x2 + x − 2, find the value of `1/alpha-1/beta`.
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Solution 1
Since 𝛼 and 𝛽 are the roots of the polynomial x + x – 2
∴ Sum of roots α + β = 1
Product of roots αβ 2 ⇒ `-1/beta`
`=(beta-alpha)/alphabeta*(alpha-beta)/alphabeta`
`=(sqrt((alpha+beta)^2-4alphabeta))/(alphabeta)`
`=sqrt(1+8)/(+2)`
`=3/2`
Solution 2
Given if α and β are the solutions of the polynomial f(x) = x2 + x − 2.
So, first let us find zeros of f(x) = 0:
The middle term x is expressed as sum of 2x and −x such that its product is equals to product of extreme terms.
(-2) x x2 = -2x2
Thus, x2 + 2x - x - 2 = 0
x(x + 2) - 1(x + 2) = 0
(x + 2)(x - 1) = 0
(x + 2) = 0 or (x - 1) = 0
=> x = -2 or x = 1
∴ α, β = (1, -2) or (-2, 1)
Case 1: When (α, β) = (1, -2)
`(1/alpha - 1/beta) = 1/1 - 1/(-2)`
= `1 + 1/2`
= `(2 + 1)/2`
∴ `1/alpha - 1/beta = (-3)/2`
Hence, `1/alpha - 1/beta = (-3)/2 or 3/2`
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