Question

find the quadratic equation sum of whose roots is 3 and sum of cubes of roots is 7

Answer 1

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