Question

√5 + 1 / √5 - 1 + √5 - 1 / √5 + 1 = a + b√5 Find the values of a and b .

Answer 1

Is your question like $\mathbf{\frac{\sqrt{5}+1}{\sqrt{5}-1}+\frac{\sqrt{5}-1}{\sqrt{5}+1}}=\mathbf{a+b\sqrt{5}}$
Okay, Let's start to solve !

Plz do rationalisation of $\mathbf{\frac{\sqrt{5}+1}{\sqrt{5}-1}}$ and $\mathbf{\frac{\sqrt{5}-1}{\sqrt{5}+1}}$ separately .
e.g., $\mathbf{\frac{\sqrt{5}+1}{\sqrt{5}-1}}$ × $\mathbf{\frac{\sqrt{5}+1}{\sqrt{5}+1}}$
= $\mathbf{\frac{(\sqrt{5}+1)^2}{\sqrt{5}^2-1}}$
= (5 + 1 + 2√5)/(5 - 1)
= (6 + 2√5)/4
= (3 + √5)/2

similarly, $\mathbf{\frac{\sqrt{5}-1}{\sqrt{5}+1}}$ × $\mathbf{\frac{\sqrt{5}-1}{\sqrt{5}-1}}$
= $\mathbf{\frac{(\sqrt{5}-1)^2}{\sqrt{5}^2-1}}$
= (5 + 1 - 2√5)/(5 - 1)
= (3 - √5)/2

Now, (3 + √5)/2 + (3 - √5)/2 = a + b√5
(3 + √5 + 3 - √5)/2 = a + b√5
3 = a + b√5
3 + 0.√5 = a + b√5
Compare both sides,
a = 3 and b = 0

Answer 2

Answer:

Step-by-step explanation:

Is your question like  

Okay, Let's start to solve !

Plz do rationalisation of  and  separately .

e.g.,  ×  

=  

= (5 + 1 + 2√5)/(5 - 1)

= (6 + 2√5)/4

= (3 + √5)/2

similarly,  ×  

=  

= (5 + 1 - 2√5)/(5 - 1)

= (3 - √5)/2

Now, (3 + √5)/2 + (3 - √5)/2 = a + b√5

(3 + √5 + 3 - √5)/2 = a + b√5

3 = a + b√5

3 + 0.√5 = a + b√5

Compare both sides,

a = 3 and b = 0

thank you