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Question
For the following, find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.
`(-3)/(2sqrt(5)), -1/2`
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Solution
Sum of the zeroes = `-3/2 sqrt(5)x`
Product of the zeroes = `- 1/2`
P(x) = x2 – (Sum of the zeroes) + (Product of the zeroes)
Then, P(x) = `x^2 - (-3/2 sqrt(5)x) - 1/2`
P(x) = `2sqrt(5)x^2 + 3x - sqrt(5)`
Using splitting the middle term method,
`2sqrt(5)x^2 + 3x - sqrt(5)` = 0
`2sqrt(5)x^2 + (5x - 2x) - sqrt(5)` = 0
`2sqrt(5)x^2 - 5x + 2x - sqrt(5)` = 0
`sqrt(5)x (2x + sqrt(5)) - (2x + sqrt(5))` = 0
`(2x + sqrt(5))(sqrt(5)x - 1)` = 0
`\implies` x = `1/sqrt(5), -sqrt(5)/2`
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