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 :

• A. \( -\frac{2}{3} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{1}{3} \)
• D. \( -\frac{1}{3} \)

Hint:

When solving for \( p \), always ensure \( p \neq 0 \) since it is the leading coefficient of a quadratic equation.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
For a quadratic polynomial \( ax^2 + bx + c \), the sum of zeroes \( \alpha + \beta = -b/a \) and the product of zeroes \( \alpha\beta = c/a \).
Step 2: Key Formula or Approach:
Given: \( f(x) = px^2 - 2x + 3p \).
Here, \( a = p \), \( b = -2 \), and \( c = 3p \).
Step 3: Detailed Explanation:
Sum of zeroes \( \alpha + \beta = -\frac{-2}{p} = \frac{2}{p} \).
Product of zeroes \( \alpha\beta = \frac{3p}{p} = 3 \).
According to the question, \( \alpha + \beta = \alpha\beta \):
\[ \frac{2}{p} = 3 \]
\[ 2 = 3p \]
\[ p = \frac{2}{3} \]
Step 4: Final Answer:
The value of \( p \) is \( \frac{2}{3} \).