A quadratic polynomial having zeroes 0 and -2, is

Question

If \( \alpha,\beta \) where \( \alpha<\beta \), are the roots of the equation \[ \lambda x^2-(\lambda+3)x+3=0 \] such that \[ \frac{1}{\alpha}-\frac{1}{\beta}=\frac{1}{3}, \] then the sum of all possible values of \( \lambda \) is:

Answer 1

Given:
Zeroes of the quadratic polynomial are \(0\) and \(-2\).

Step 1: Construct the polynomial from zeroes
If zeroes are \(\alpha\) and \(\beta\), the polynomial is:
\[k(x - \alpha)(x - \beta)\] With \(\alpha = 0\) and \(\beta = -2\), we have:
\[k(x - 0)(x - (-2)) = kx(x + 2)\]

Step 2: Determine the constant \(k\)
The polynomial is \(4x(x + 2)\), thus \(k = 4\).

Final Answer:
\[\boxed{4x(x + 2)}\]

A quadratic polynomial having zeroes 0 and -2, is

The Correct Option is B

Solution and Explanation

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