(i) 1/4, -1 (ii) √2, 1/3 (ii) 0, √5 (iv) 1, 1 (v) -1/4, 1/4 (vi) 4, 1

**Solution:**

**(i) 1/4, -1 **

From the formulas of sum and product of zeroes, we know,

Sum of zeroes = α + β

Product of zeroes = α β

Sum of zeroes = α + β = 1/4

Product of zeroes = α β = -1

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2 }– (α + β)x + αβ = 0

x^{2 }– (1/4)x + (-1) = 0

4x^{2 }– x – 4 = 0

Thus**, **4x^{2 }– x – 4 is the quadratic polynomial.

**(ii) √2, 1/3**

Sum of zeroes = α + β =√2

Product of zeroes = α β = 1/3

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2 }– (α + β)x + αβ = 0

x^{2} –(√2)x + (1/3) = 0

3x^{2 }– 3√2x + 1 = 0

Thus, 3x^{2 }– 3√2x + 1 is the quadratic polynomial.

**(iii) 0, √5**

Given,

Sum of zeroes = α + β = 0

Product of zeroes = α β = √5

∴ If α and β are zeroes of any quadratic polynomial, then the quadratic polynomial equation can be written directly as:-

x^{2 }– (α + β)x + αβ = 0

x^{2 }– (0)x + √5 = 0

Thus, x^{2 }+ √5 is the quadratic polynomial.

**(iv) 1, 1**

Given,

Sum of zeroes = α + β = 1

Product of zeroes = α β = 1

x^{2 }– (α + β)x + αβ = 0

x^{2 }– x + 1 = 0

Thus, x^{2 }– x + 1 is the quadratic polynomial.

**(v) -1/4, 1/4**

Given,

Sum of zeroes = α+β = -1/4

Product of zeroes = α β = 1/4

x^{2}–(α+β)x +αβ = 0

x^{2}–(-1/4)x +(1/4) = 0

4x^{2}+x+1 = 0

Thus, 4x^{2}+x+1 is the quadratic polynomial.

**(vi) 4, 1**

Given,

Sum of zeroes = α + β = 4

Product of zeroes = αβ = 1

x^{2 }– (α + β)x + αβ = 0

x^{2 }– 4x + 1 = 0

Thus, x^{2 }– 4x + 1 is the quadratic polynomial.