Question

If \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 5x + k = 0 \), and \( 4(\alpha^2 + \beta^2 + \alpha\beta) = 23 \), then which of the following is true?

• A. \( k^2 - 3k + 2 = 0 \)
• B. \( k^2 + 3k - 2 = 0 \)
• C. \( k^2 - 2k + 3 = 0 \)
• D. \( k^2 - 2k - 3 = 0 \)

Hint:

When dealing with quadratic equations and sums of roots, use Vieta’s formulas to find relations between the coefficients and the roots.

Answer 1

The Correct Option is A

Given that \( \alpha \) and \( \beta \) are the roots of the equation \( 2x^2 + 5x + k = 0 \), we can use Vieta's formulas:
- \( \alpha + \beta = -\frac{5}{2} \)
- \( \alpha\beta = \frac{k}{2} \)
We are given that \( 4(\alpha^2 + \beta^2 + \alpha\beta) = 23 \).
Using the identity \( \alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta \), we substitute the values: \[ 4\left(\left(\frac{5}{2}\right)^2 - 2 \cdot \frac{k}{2} + \frac{k}{2}\right) = 23 \] Simplifying the equation will lead to \( k^2 - 3k + 2 = 0 \).