find the quadratic equation sum of whose roots is 3 and sum of cubes of roots is 7
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Let the roots be α and β
The sum of the roots is 3:
α + β = 3 ------------------[ 1 ]
The sum of the cubes of the roots is 7:
α³ + β³ = 7
(α + β)³ - 3αβ(α + β) = 7 ------------------[ 2 ]
Sub [ 1 ] into [ 2 ]:
(3)³ - 3αβ(3) = 7
27 - 9αβ = 7
9αβ = 20
αβ = 20/9
So now we know that:
Sum of the roots = α + β = 3
Product of the roots = αβ = 20/9
Form the quadratic equation:
x² - (α + β) x + αβ = 0
x² - 3x + 20/9 = 0
9x² -27x + 20 = 0
Answer: 9x² -27x + 20 = 0
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