Question

If \(a=\frac{\sqrt5+1}{\sqrt5-1}\) and \(b=\frac{\sqrt5-1}{\sqrt5+1}\), then \(a^2+ab+b^2=\)

• A. 9
• B. 8
• C. 6
• D. 5

Hint:

Remember the identity \[ a^2+ab+b^2=(a+b)^2-ab. \] It often simplifies irrational-expression problems quickly.

Answer 1

The Correct Option is B

Concept: Use algebraic identities involving \(a+b\) and \(ab\).

Step 1: Finding \(ab\).
\[ ab= \frac{\sqrt5+1}{\sqrt5-1} \cdot \frac{\sqrt5-1}{\sqrt5+1} =1 \]

Step 2: Finding \(a+b\).
\[ a+b= \frac{(\sqrt5+1)^2+(\sqrt5-1)^2}{5-1} \] \[ =\frac{(6+2\sqrt5)+(6-2\sqrt5)}{4} =\frac{12}{4} =3 \]

Step 3: Evaluating the expression.
\[ a^2+ab+b^2=(a+b)^2-ab \] \[ =3^2-1 \] \[ =9-1 \] \[ =8 \] {8}