Question

If x+y=5 and xy=3

`√(x ) + √(y = )`
​

Answer 1

Notice that if both
`x,y` are positive, then

`xy = 3 \implies √(xy) = \sqrt x \sqrt y = \sqrt3`

We also have the binomial expansion

`x + 2√(x)√(y) + y = \left(\sqrt x\right)^2 + 2 \sqrt x \sqrt y + \left(\sqrt y\right)^2 = \left(\sqrt x + \sqrt y\right)^2`

Then

`\left(\sqrt x + \sqrt y\right)^2 = (x + y) + 2√(xy) = 5 + 2\sqrt3 \\\\ \implies \sqrt x + \sqrt y = \boxed{√(5 + 2\sqrt3)}`

Let's see if we can denest the radical. Suppose we could write

`√(5 + 2\sqrt3) = a + b \sqrt c`

for some non-zero integer constants
`a,b,c` (and
`c>0`). Squaring both sides gives

`5 + 2\sqrt3 = a^2 + 2ab\sqrt c + b^2c`

Let
`c=3` and
`ab=1`. Then
`a^2 + b^2c = 5`, which gives

`a^2 + 3b^2 = 5`

`a^2 + \frac3{a^2} = 5`

`a^4 - 5a^2 + 3 = 0`

We can solve this with the quadratic formula. However, it'll lead to non-integer solutions for
`a,b`, so we cannot denest after all.