Question

If $a = \sqrt{5} - \sqrt{3}$, then the value of $a^2 + 2a\sqrt{3} + 8$ is:

• A. 10
• B. 12
• C. 15
• D. 18

Hint:

Use $(a-b)^2 = a^2 + b^2 - 2ab$ and simplify step by step.

Answer 1

The Correct Option is A

Concept: Expand and simplify using identities.
Step 1: Find $a^2$.
\[ a^2 = (\sqrt{5} - \sqrt{3})^2 = 5 + 3 - 2\sqrt{15} = 8 - 2\sqrt{15} \]
Step 2: Find $2a\sqrt{3}$.
\[ 2a\sqrt{3} = 2(\sqrt{5} - \sqrt{3})\sqrt{3} = 2(\sqrt{15} - 3) \]
Step 3: Add all terms.
\[ a^2 + 2a\sqrt{3} + 8 \] \[ = (8 - 2\sqrt{15}) + (2\sqrt{15} - 6) + 8 \] \[ = 8 - 6 + 8 = 10 \]
Hence, the value is 10.