If α + β = 5 and α^{3} +β^{3} = 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`