If $\alpha,\ \beta$ are the zeroes of $f( x)=px^2-2x+3p$ and $\alpha +\beta=\alpha\beta$, then find the value of $p$.
Given: $\alpha , \ \beta$ are zeroes of the polynomial $f( x)=px^2−2x+3p$, and $\alpha+\beta =\alpha \beta$
To do: To find the value of $p$.
Solution: From the given quadratic equation,
$\alpha +\beta=\frac{2}{p}$ and $\alpha \beta =3$
but it is given,
$\alpha +\beta =\alpha \beta$
$\Rightarrow \frac{2}{p}=3$
$\Rightarrow p=\frac{2}{3}$
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