Question

If alpha +beta =3 and alpha ^3+beta^3 =7 then alpha and beta are the roots of the equation

Answer 1

Given α+β=3
     cubing on both sides
  (α+β)³=3³
α³+β³+3αβ(α+β)=27
7+3αβ(3)=27
9αβ=20
αβ=20/9
now,
α+β=3
α=3-β 

substitute in αβ=20/9
(3-β)β=20/9
3β-β²=20/9
27β-9β²=20
9β²-27β+20=0
by solving we get β=5/3,4/3
substitute the values in α+β
we get α=5/3,4/3

substitute α andβ values in (x-α)(x-β) 
let α=5/3 and β=4/3
we get,
9x²-27x+20.

Answer 2

Answer:

Step-by-step explanation:

Given α+β=3

     cubing on both sides

  (α+β)³=3³

α³+β³+3αβ(α+β)=27

7+3αβ(3)=27

9αβ=20

αβ=20/9

now,

α+β=3

α=3-β 

substitute in αβ=20/9

(3-β)β=20/9

3β-β²=20/9

27β-9β²=20

9β²-27β+20=0

by solving we get β=5/3,4/3

substitute the values in α+β

we get α=5/3,4/3

substitute α andβ values in (x-α)(x-β) 

let α=5/3 and β=4/3

we get,

9x²-27x+20

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