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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`
Solution
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
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