If the difference of the roots of the quadratic equation is 5 and the difference of their cubes is 215, find the quadratic equation.

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