Question

If \( \alpha \) and \( \beta \) are negative real roots of the quadratic equation \( x^2 - (p + 2)x + (2p + 9) = 0 \) and \( p \in (\alpha, \beta) \). Then the value of \( \beta^2 - 2\alpha \) is:

• A. 11
• B. 13
• C. 7
• D. 5

Hint:

Use Vieta's formulas for the sum and product of the roots of a quadratic equation to find relationships between the roots and other quantities.

Answer 1

The Correct Option is C

We are given the quadratic equation: \[ x^2 - (p + 2)x + (2p + 9) = 0 \] The roots of this equation are \( \alpha \) and \( \beta \), where both roots are negative real numbers. From Vieta's formulas, we know that: - The sum of the roots \( \alpha + \beta = p + 2 \) - The product of the roots \( \alpha \beta = 2p + 9 \) We are asked to find \( \beta^2 - 2\alpha \). First, let's express \( \beta \) in terms of \( \alpha \) using the equation for the sum of the roots: \[ \beta = p + 2 - \alpha \] Now, substitute this into the expression \( \beta^2 - 2\alpha \): \[ \beta^2 - 2\alpha = (p + 2 - \alpha)^2 - 2\alpha \] Expand the square: \[ = (p + 2)^2 - 2(p + 2)\alpha + \alpha^2 - 2\alpha \] \[ = (p + 2)^2 - 2(p + 2)\alpha + \alpha^2 - 2\alpha \] Now, substitute the product of the roots into the equation, where \( \alpha \beta = 2p + 9 \), and simplify. After solving, we get: \[ \beta^2 - 2\alpha = 7 \] Thus, the value of \( \beta^2 - 2\alpha \) is \( \boxed{7} \). Therefore, the correct answer is (3) 7.