Question

If α and β are the roots of the quadratic equation `x^2 + sqrt(2)x + 3` = 0, form a quadratic polynomial with zeroes `1/α, 1/β`

Answer 1

Given α and β are the roots of the quadratic polynomial

`x^2 + sqrt(2)x + 3` = 0   ......(1)

Sum of the roots α + β = `-"b"/"a"`

α + β = `- sqrt(2)/1`

α + β = `- sqrt(2)`

Product of the roots α  β = `"c"/"a"`
α  β = `3/1`

α  β = 3

`1/α + 1/β = (β+ α)/(α  β) = (α + β)/(α  β)`

`1/α + 1/β = - sqrt(2)/3`

`1/α xx 1/β = 1/(αβ)= 1/3`

∴ The required quadratic equation whose roots are `1/α, 1/β` is

x² – (sum of the roots)x + product of the roots = 0

`x^2 - (1/α +1/β)x + (1/α * 1/β)` = 0

`x^2 - (- sqrt(2)/3)x + 1/3` = 0

`x^2 + sqrt(2)/3x + 1/3` = 0

`3x^2 + sqrt(2)x + 1` = 0