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`

Answer 1

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`