If $\alpha$ and $\beta$ are two zeroes of a polynomial $f(x) = px^2...

Question

If $\alpha$ and $\beta$ are two zeroes of a polynomial $f(x) = px^2 - 2x + 3p$ and $\alpha + \beta = \alpha\beta$, then value of $p$ is :

Accepted Answer

**Step 1 — Write the relation between zeroes and coefficients.**
For $ax^2+bx+c$ : $\ \alpha+\beta = -\dfrac{b}{a}$ and $\alpha\beta = \dfrac{c}{a}$.

Here $a = p$, $b = -2$, $c = 3p$.

**Step 2 — Substitute the values.**
$$\alpha+\beta = -\frac{-2}{p} = \frac{2}{p}, \qquad \alpha\beta = \frac{3p}{p} = 3$$

**Step 3 — Apply the given condition $\alpha+\beta = \alpha\beta$.**
$$\frac{2}{p} = 3$$

**Step 4 — Solve for $p$.**
$$2 = 3p \ \Rightarrow\ p = \frac{2}{3}$$

**Answer : (B) $\dfrac{2}{3}$**