Question

If \( x = \sqrt{5} + \sqrt{2} \), \( y = \sqrt{5} - \sqrt{2} \), then \[ 3x^2 + 4xy - 3y^2 \quad \text{is equal to:} \]

• A. \( \frac{1}{3} \left( 56\sqrt{10} - 12 \right) \)
• B. \( \frac{1}{3} \left( 56\sqrt{10} + 12 \right) \)
• C. \( \frac{1}{3} \left( 56\sqrt{10} - 10 \right) \)
• D. None of these

Hint:

To simplify expressions with square roots, first expand and then collect like terms.

Answer 1

The Correct Option is A

Step 1: Expand the given expression.
We are given \( x = \sqrt{5} + \sqrt{2} \) and \( y = \sqrt{5} - \sqrt{2} \), and need to calculate the value of \( 3x^2 + 4xy - 3y^2 \). First, calculate \( x^2 \), \( y^2 \), and \( xy \): \[ x^2 = (\sqrt{5} + \sqrt{2})^2 = 5 + 2\sqrt{10} + 2 = 7 + 2\sqrt{10} \] \[ y^2 = (\sqrt{5} - \sqrt{2})^2 = 5 - 2\sqrt{10} + 2 = 7 - 2\sqrt{10} \] \[ xy = (\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2}) = 5 - 2 = 3 \]

Step 2: Substitute into the expression.
Now substitute these into the original expression:
\[ 3x^2 + 4xy - 3y^2 = 3(7 + 2\sqrt{10}) + 4(3) - 3(7 - 2\sqrt{10}) \] Simplifying: \[ = 3 \times 7 + 3 \times 2\sqrt{10} + 12 - 3 \times 7 + 3 \times 2\sqrt{10} \] \[ = 21 + 6\sqrt{10} + 12 - 21 + 6\sqrt{10} \] \[ = 12 + 12\sqrt{10} \]

Step 3: Final simplification.
Now, factor out \( \frac{1}{3} \) from the expression: \[ \frac{1}{3} \left( 56\sqrt{10} - 12 \right) \] Thus, the correct answer is \( \frac{1}{3} \left( 56\sqrt{10} - 12 \right) \), which corresponds to option (A).