# If α + β = 5 and α^3 +β^3 = 35, find the quadratic equation whose roots are α and β. - Algebra

If α + β = 5 and α33 = 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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