Find the quadratic polynomial if the sum and product of its zeroes are $\sqrt{3},-1$.

Given:
Sum of zeroes, $\mathrm{\alpha }+\mathrm{\beta }=\sqrt{3}$,
Product of zeroes ,$\mathrm{\alpha \beta }=-1$

Now, the quadratic polynomial can be given as,

${\mathbf{x}}^{\mathbf{2}}\mathbf{-}\mathbf{\left(}\mathbf{\text{ Sum of root }}\mathbf{\right)}\mathbf{x}\mathbf{+}\mathbf{\left(}\mathbf{\text{ Product of root }}\mathbf{\right)}\mathbf{=}\mathbf{0}$

${\mathrm{x}}^2-(\mathrm{\alpha }+\mathrm{\beta })\mathrm{x}+\left(\mathrm{\alpha \beta }\right)=0$
Put the values,
${\mathbf{x}}^{\mathbf{2}}\mathbf{-}\sqrt{\mathbf{3}}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{0}$

Hence, the quadratic polynomial is: ${\mathbf{x}}^{\mathbf{2}}\mathbf{-}\sqrt{\mathbf{3}}\mathbf{x}\mathbf{-}\mathbf{1}\mathbf{=}\mathbf{0}$