**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 α

^{3}+ β^{3}= 63 ............ eq. (2)
We know that,

α

^{3}+ β^{3}= (α + β )^{3}– 3αβ (α + β )
∴ 63 = 3

^{3}– 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,

x

^{2}– (Sum of the roots)x + Product of the roots = 0
∴ x

^{2}–( α + β)x + αβ = 0
∴ x

^{2}– 3x + (-4) = 0
∴ x

^{2}– 3x – 4 = 0