How do you simplify the following: (sqrt(6) + sqrt(2))/2

Not all expressions of this form can be simplified; but if one can, we would like the square root to have the form sqrt(a) + sqrt(b) or sqrt(a) - sqrt(b) where a and b are positive rational numbers, and a > b to give the positive square root. If we assume that sqrt(2-sqrt(3)) has this form, then this must be true: (sqrt(a) +- sqrt(b))^2 = 2 - sqrt(3) Expanding, we get a + b +- 2 sqrt(ab) = 2 - sqrt(3) Equating the rational and irrational parts of this equation, a + b = 2 +-2 sqrt(ab) = -sqrt(3) The second equation here tells us that we will have to choose the negative sign in order to have a solution, since square roots have to be positive. If we now square this equation, we have a + b = 2 4ab = 3 Solving these equations by substituting in the second, we get 4a(2 - a) = 3 4a^2 - 8a - 3 = 0 8 +- sqrt(64 + 48) a = = 1/2 or 3/2 8 and b = 2 - a = 3/2 or 1/2. One choice here will give the positive root, the other the negative root. We take a > b, so that a = 3/2 and b = 1/2. Consequently, our solution is sqrt(2 - sqrt(3)) = sqrt(3/2) - sqrt(1/2) = (sqrt(6) - sqrt(2))/2 Looking for information on this technique, which I didn't recall being taught, I found this page, which together with some rather advanced techniques includes a quote from an old algebra text about the method:

Okay so for every triangle has three angles that always add up to 180 degrees. and to tell the difference between whether they are secure, right or obtuse is that acute: has one angle that is less than 90 degrees. right triangle means there is at least one angle that is exactly 90 degrees. and an obtuse angle has an angle that is bigger than 180 degrees. i’ll do #13 as an example: 13. 40 and 50 so if every triangle has 180 degrees 40+50= 90 degrees then180-90= 90 degrees so the third angle is 90 degrees and because this triangle has a 90 degree angle it means it is a right triangle