If a quadratic equation 4x2 + 5x – 1 = 0 have two roots α and β, then what will be the equation, whose roots are (α2 + β2) and (α2β + β2α)?

If a quadratic equation 4x2 + 5x – 1 = 0 have two roots α and β, then what will be the equation, whose roots are (α2 + β2) and (α2β + β2α)? Correct Answer 256x² – 608x + 165 = 0

GIVEN:

Quadratic equation 4x² + 5x – 1 = 0 have two roots α and β

FORMULA USED:

(α + β) = -b/a and (αβ) = c/a

CALCULATION:

Given equation is

4x² + 5x – 1 = 0

So, (α + β) = -5/4 and αβ = -1/4

Now, α’ = (α² + β²)

⇒ α' = (α + β)² - 2αβ

⇒ α’ = (25/16) + (1/2)

⇒ α’ = 33/16

and β’ = (α²β + β²α)

⇒ β’ = αβ(α + β) = (-1/4)(-5/4)

⇒ β’ = 5/16

The new equation will be-

x² – (α’ + β’)x + α’β’ = 0

⇒ x² – + (33/16)(5/16) = 0

⇒ x² – (19x/8) + (165/256) = 0

⇒ 256x² – 608x + 165 = 0