Question

\( \theta \) and \( \gamma \) are the roots of the equation \( x^2 - \alpha x + \beta = 0 \) and if \( \theta + \gamma = \alpha \), then what is the value of \( \theta^2 + \gamma^2 \)?

• A. \( \alpha^2 - 2\beta \)
• B. \( \alpha^2 + 2\beta \)
• C. \( \alpha^2 - 4\beta \)
• D. \( \alpha^2 + 4\beta \)

Hint:

To find \( \theta^2 + \gamma^2 \), use the identity \( \theta^2 + \gamma^2 = (\theta + \gamma)^2 - 2\theta \gamma \).

Answer 1

The Correct Option is C

Step 1: Using the given information.
We know that the sum of the roots \( \theta + \gamma = \alpha \) and the product of the roots \( \theta \gamma = \beta \). Using the identity \( \theta^2 + \gamma^2 = (\theta + \gamma)^2 - 2\theta \gamma \), we can substitute the values for \( \theta + \gamma \) and \( \theta \gamma \).

Step 2: Conclusion.
The value of \( \theta^2 + \gamma^2 \) is \( \alpha^2 - 4\beta \), corresponding to option (3).