Question

if the difference of the zeros of the quadratic equation is 5 and the difference of their cubes is 215 find the quadratic equation​

Answer 1

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

Answer 2

Please see the attached answer