Question

the square root of 3+√5 is​

Answer 1

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Let

√ 3+√5 = √x + √y

squaring in both sides

( √ 3+√5 ) ( √ 3+√5 )

= ( √x + √y ) ( √x + √y )

3 + √5 = x + y + 2√xy

It implies that

3 = x + y and √5 = 2√xy

Squaring the later equation

5 = 4xt

from this

y = 5/4x

substitute the right hand side into

x + y = 3 to get x + 5/4x = 3

multiply by 4x

4x ( x ) + 4x ( 5/4x ) = 4x ( 3 )

4x² + 5 = 12x

4x² - 12x + 5 = 0

( 2x - 5 ) ( 2x - 1 ) = 0

x = 5/2 or x = 1/2

Now

x = 5/2

means that

y = 5/ 4(5/2)

y = 5/2(5)

y = 5/10 = 1/2

so

x = 5/2 and y = 1/2

or

x = 1/2 and y = 5/2

so x and y are interchangeable

:. √ 3+√5 = √1/2 + √5/2