4. If the
difference of the roots of the quadratic equation is 5 and the difference of
their cubes is 215, find the quadratic equation.
Sol.
Let α and β be the roots of a quadratic equation.
∴ α - β = 5 [Given] ...... eq. (1)
and α3 - β3 = 215 ............ eq. (2)
We know that,
α3 - β3
= (α - β )3 + 3αβ (α - β )
∴ 215 = 53 + 3αβ
(5) [ From eq. (1) & (2)]
∴ 215 – 125 = 15αβ
∴ 90 = 15αβ
∴ α β = 90/15
∴ α β = 6
Also, (α - β )2 =
(α + β )2 – 4αβ
∴ 52 = (α + β )2
– 4(6)
∴ 25 + 24 = (α + β )2
∴ (α + β )2 = 49
∴ α + β = ± √49
∴ α + β = ± 7
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 – (±7)x + (6)
= 0
∴ x2 ±7x +6 = 0
