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Question
If α + β = 5 and α3 +β3 = 35, find the quadratic equation whose roots are α and β.
Solution 1
`alpha+beta=5 ......(1)`
`alpha^3+beta^3=35......(2)`
`alpha^3+beta^3=(alpha+beta)^3-3alphabeta(alpha+beta)`
`=(5)^3-3alphabeta(5)`
`=125-15alphabeta`
`therefore 125-15alphabeta=35 `
`therefore 15alphabeta=90`
`thereforealphabeta=6`
The required quadratic equation is
`x^2-(alpha+beta)x+alphabeta=0`
`x^2-5x+6=0`
Solution 2
Here α and β are the roots of the quadratic equation, so required equations is
`x^2 - (alpha + beta)x + alphabeta` ....(1)
We have `alpha+beta = 5` and `alpha^3 + beta^3 = 35`
`alpha^3 + beta^3 = (alpha + beta)^3 - 3alphabeta (alpha+ beta)`
`:. 35 = (5)^3 - 3alphabeta xx 5`
`:. 35 = 125 - 15alphabeta`
`:. 15alphabeta = 90`
`:. alphabeta = 6`
So from (1) required quadratic equation is `x^2 - 5x + 6 = 0`
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