If the sum of the roots of the quadratic is 3 and sum of their cubes is 63, find the quadratic equation.

3. If the sum of the roots of the quadratic is 3 and sum of their cubes is 63, find the quadratic equation.

Sol. Let α  and β  be the roots of a quadratic equation.

∴  α  + β  = 3 [Given] ...... eq. (1)

and α³ + β³ = 63  ............ eq. (2)

We know that,      

α³ + β³ = (α + β )³ – 3αβ (α + β )

∴ 63 = 3³ – 3αβ (3)   [ From eq. (1) & (2)]

∴ 63 – 27 = - 9αβ

∴ 36 = - 9αβ

∴ α β = - 36/9

   ∴ α β = - 4

We know that, Quadratic equation is given by,

x² – (Sum of the roots)x + Product of the roots = 0

∴ x² –( α  + β)x + αβ = 0

∴ x² – 3x + (-4) = 0

∴ x² – 3x – 4 = 0