Question

find square root of 3+√5​

Answer 1

Answer:

What is the square root of 3+root 5?

This is a problem from 2018 that was in my feed in July 2019.

Because

3>5–√,

the radicals can be unnested.

Let

3+5–√−−−−−−√=x−−√+y√.

Square both sides and get

(3+5–√−−−−−−√)(3+5–√−−−−−−√)

=(x−−√+y√)(x−−√+y√).

This means that

3+5–√=x+y+2xy−−√.

It implies that

3=x+y and 5–√=2xy−−√.

Square the latter equation and get

5=4xy.

From this

y=54x.

Substitute the right-hand side into

x+y=3

to get

x+54x=3.

Multiply by 4x:

4x(x)+4x(54x)=4x(3),

4x2+5=12x,

4x2−12x+5=0,

(2x−5)(2x−1)=0.

Then

x=52 or x=12.

Now

x=52

means that

y=54(52)=52(5)=12.

And then

x=12

means that

y=54(12)=52.

So either

x=52 and y=12

or

x=12 and y=52.

So x and y are interchangeable. This is because

3=x+y and 5=4xy

are symmetric in x and y.

∴3+5–√−−−−−−√=12−−√+52−−√.

Step-by-step explanation:

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