**Q) **If 𝛼, β are zeroes of quadratic polynomial 5x^{2} + 5x + 1, find the value of

1. 𝛼^{2} + 𝛽^{2}

2. 𝛼‾^{1} + 𝛽‾^{1}

**Ans: **Given polynomial equation 5x^{2} + 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 𝛼 ^{2} + 𝛽^{2:}**

𝛼^{2} + 𝛽^{2 }= (𝛼 + 𝛽)^{2 } – 2 (α . β)

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

𝛼^{2} + 𝛽^{2 }= (-1)^{2 } – 2 )

= 1 – =

**The value of 𝛼 ^{2} + 𝛽^{2} 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.**