Q)  If 𝛼, β are zeroes of quadratic polynomial 5x² + 5x + 1, find the value of
1. 𝛼² + 𝛽²
2. 𝛼‾¹ + 𝛽‾¹

Ans: Given polynomial equation 5x² + 5x + 1 = 0

The roots of the polynomial be α and β.

We know that sum of roots (α + β) = 

   α + β =  = -1

Hence, α = -1 – β …………… (i)

Next, we know that the product of the roots (α x β) =

α . β = …………. (ii)

(1) Value of 𝛼² + 𝛽^2:

𝛼² + 𝛽² = (𝛼 + 𝛽)²  – 2 (α . β)

By transferring values from equations (i) and (ii), we get:

𝛼² + 𝛽² = (-1)²  – 2 )

= 1 –   =

The value of 𝛼² + 𝛽² is

(2) Value of 𝛼^-1 + 𝛽^-1

𝛼^-1 + 𝛽^-1  =

=

By transferring values from equations (i) and (ii), we get:

𝛼^-1 + 𝛽^-1  = = – 5

Therefore, the Value of 𝛼^-1 + 𝛽^-1 is – 5.