Question

If $\alpha$ and $\beta$ are the zeroes of a polynomial $x^2-4\sqrt{3}x+3$, then find the value of $\alpha +\beta -\alpha \beta$.

Answer 1

We know that for any quadratic polynomial

$p\left(x\right)=a{x}^2+bx+c$ where $a\ne 0$

Let $\alpha ,\beta$ be the zeroes of the polynomial $p\left(x\right)$

Then, $\alpha +\beta =\frac{-b}{a}$ and $\alpha \beta =\frac{c}{a}$

Now, given that

$p\left(x\right)=x^2-4\sqrt{3}x+3$

So, we have $\alpha +\beta =4\sqrt{3}$

$\alpha \beta =3$

Using above values we get

$\alpha +\beta$ $-\left(\alpha \beta \right)=4\sqrt{3}-3$