Question

Find a quadratic polynomial whose sum and product respectively of the zeroes are as given. Also find the zeroes of these polynomials by factorisation.

$-2\sqrt{3},-9$

Answer 1

As per question,

$\alpha +\beta =-2\sqrt{3}$

$\alpha \times \beta =-9$

$\alpha =\frac{-9}{\beta }$

The quadratic polynomial,

$x^2-$(sum of the roots)$x$ + (product of the roots) = 0

$x^2-(-2\sqrt{3})x-9=0$

$x^2+\left(2\sqrt{3}\right)x-9=0$

$\alpha +\beta =-2\sqrt{3}$

$\alpha +\frac{-9}{\alpha }=-2\sqrt{3}$

$\frac{{\alpha }^2-9}{\alpha }=-2\sqrt{3}$

${\alpha }^2+2\sqrt{3}\alpha -9=0$

${\alpha }^2+3\sqrt{3}\alpha -\sqrt{3}\alpha -9=0$

$(\alpha +3\sqrt{3})(\alpha -\sqrt{3})=0$

$\alpha ,\beta =-3\sqrt{3},\sqrt{3}$