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:

For any quadratic polynomial \(ax^2 + bx + c\), if the sum of roots equals the product of roots, then we can directly use the relation:
\[ -b = c \]
Here, with \(b = -2\) and \(c = 3p\):
\[ -(-2) = 3p \implies 2 = 3p \implies p = \frac{2}{3} \]
Using this direct shortcut bypasses the need to write out fractions and saves valuable time!

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a quadratic polynomial \(f(x) = px^2 - 2x + 3p\) with zeroes \(\alpha\) and \(\beta\). We are also given the condition that the sum of the zeroes is equal to the product of the zeroes: \(\alpha + \beta = \alpha\beta\). We need to find the value of the parameter \(p\).

Step 2: Key Formula or Approach:
For a standard quadratic polynomial \(ax^2 + bx + c\), the relationship between its coefficients and its zeroes \(\alpha\) and \(\beta\) are given by Vieta's formulas:
Sum of zeroes:
\[ \alpha + \beta = -\frac{b}{a} \]
Product of zeroes:
\[ \alpha\beta = \frac{c}{a} \]

Step 3: Detailed Explanation:
1. Identify the coefficients of the given polynomial \(f(x) = px^2 - 2x + 3p\) by comparing it with \(ax^2 + bx + c\):
\[ a = p, \quad b = -2, \quad c = 3p \]
2. Express the sum of the zeroes algebraically:
\[ \alpha + \beta = -\frac{b}{a} = -\frac{-2}{p} = \frac{2}{p} \]
3. Express the product of the zeroes algebraically:
\[ \alpha\beta = \frac{c}{a} = \frac{3p}{p} \]
Since \(p \neq 0\) (as it is the leading coefficient of a quadratic polynomial), we can cancel \(p\) from the numerator and denominator:
\[ \alpha\beta = 3 \]
4. Apply the given condition \(\alpha + \beta = \alpha\beta\):
\[ \frac{2}{p} = 3 \]
5. Solve this linear equation for \(p\):
\[ 3p = 2 \implies p = \frac{2}{3} \]
This gives the value of \(p\) as \(\frac{2}{3}\).

Step 4: Final Answer:
The value of \(p\) is \(\frac{2}{3}\), which corresponds to option (B).