Question

√5 + 1 =a+b√ 5
√5 -1​

Answer 1

a = 3/2 , b = 1/2

Given:

$\dfrac{\sqrt{5} + 1 }{ \sqrt{5} - 1 } = a + b \sqrt{5}$

To Find:

The value of a and b

What you have to do ?

1. Rationalize the denominator.

2. Use Algebra Formula.

3. Simplify and Get the answer.

Solution:

L.H.S

$\frac{( \sqrt{5} + 1)( \sqrt{5} + 1) }{( \sqrt{5} - 1)( \sqrt{5} + 1) } \\ \\ = \frac{ { (\sqrt{5})}^{2} + {1}^{2} + 2. \sqrt{5} .1 }{ { (\sqrt{5} )}^{2} - {(1)}^{2} } \\ \\ = \frac{5 + 1 + 2 \sqrt{5} }{5 - 1} \\ \\ = \frac{6 + 2 \sqrt{5} }{4} \\ \\ = \frac{2(3 + \sqrt{5})}{4} \\ \\ = \frac{3}{2} + \frac{ \sqrt{5} }{2}$

Now, Comparing the L.H.S from R.H.S I get,

$a = \frac{3}{2}\\ \\ b \sqrt{5}= \frac{\sqrt{5} }{2} \implies b = \frac{1}{2}$

Algebra Formula Used

(a+b)(a+b) =a² + b² + 2ab

(a+b)(a-b) = a²- b²