How do you simplify √(5 + 2√6)

can someone do a step by step explanation for me? I find this hard to understand...

also for questions like √(9 - 4√5) and √(4 + √15)

i) Let √(5 + 2√6) = √x + √y

ii) Squaring both sides, 5 + 2√6 = x + y + 2√(xy)

iii) Equating the rational and irrational parts on both sides,

x + y = 5 and xy = 6

iv) Applying, (x - y) = √{(x + y)² - 4xy}

(x - y) = 1

v) Thus solving (x + y) = 5 and (x - y) = 1, x = 3 and y = 2

So, √(5 + 2√6) = √3 + √2

Verify: Squaring right side,it is: 3 + 2√6 + 2 = 5 + 2√6.

Q.2:

i) In similar process as above, let √(9 - 4√5) = √x - √y [Here we adopt √x - √, since in the given one we have negative sign].

ii) Squaring and equating the rational and irrational parts.

x + y = 9 and xy = 20; since here 2√(xy) = 4√5.

As in preious here also we get (x - y) = 1

Solving, x = 5 and y = 4

Hence square root = √5 - 2

Q.3:

This is similar to the first one; you may solve yourself by keeping it as √x + √y.

Ans: √(5/2) + √(3/2)

... √[ 5 + 2√6 ]

= √[ 3 + 2 √((3)(2)) ]

= √[ (√3)² + (√2)² + 2√(3).√(2) ]

= √[ (√3) + (√2) ]²

= (√3) + (√(2) ........................................ Ans.
___________________________

... 9 - 4√5

= 5 + 4 - 2(2)√5

= (√5)² + 2² - 2(2)(√5)

= [ (√5) - 2 ]²

√(9-4√5) = (√5) - 2 .................................. Ans.
____________________________

... 4 + √(15)

= (1/2)[ 8 + 2√(15) ]

= (1/2) [ 5 + 3 + 2√((5)(3)) ], ...
_____________________________

I know of no way to simplify those. Any of those.