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
