Relation Between Zeroes (Roots) and Coefficients of a Quadratic Equation

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`4x^2 + 5sqrt(2)x - 3`

Find the zeroes of the following polynomials by factorisation method and verify the relations between the zeroes and the coefficients of the polynomials:

`2x^2 + (7/2)x + 3/4`

If α and β are the zeros of the quadratic polynomial f(x) = x² − 1, find a quadratic polynomial whose zeroes are `(2alpha)/beta" and "(2beta)/alpha`

If one zero of the polynomial p(x) = 6x² + 37x – (k – 2) is reciprocal of the other, then find the value of k.

Find a cubic polynomial with the sum, sum of the product of its zeroes taken two at a time, and the product of its zeroes as 2, − 7, − 14 respectively

Find the quadratic polynomial, sum of whose zeroes is `sqrt2` and their product is `(1/3)`. 

Find a quadratic polynomial whose zeroes are 6 and – 3.

If α, β, γ are are the zeros of the polynomial f(x) = x³ − px² + qx − r, the\[\frac{1}{\alpha\beta} + \frac{1}{\beta\gamma} + \frac{1}{\gamma\alpha} =\]