Can You Simplify Another Radical?

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PLAYLISTS 🎵 :

[petrileskinen]

Interesting, instead of a constant 9 this could be applied for a more general case sqrt(a-sqrt(a^2 - b^2)) = sqrt((a+b)/2) - sqrt((a-b)/2)

[yurenchu]

8:43 Note: these happen to be the same values, but note that x = (a-3)/2 is *not* a solution to the original problem. It's a so-called _extraneous_ solution, introduced by squaring the equation at 3:30 .

[-wx--]

Apply nested radical formula: √(x±√y) = √[½(x+D)] ± √[½(x−D)], where D = √(x²−y).
D = √[a²−(a²−9)] = √9 = 3,
√[a−√(a²−9)] = √[½(a+3)] − √[½(a−3)].
It works every time x²−y is a perfect square:
√[3−√(9−a²)] = √[½(3+|a|)] − √[½(3−|a|)], since D = √[3²−(9−a²)] = |a|.

[neuralwarp]

You forgot early to address the negative square root of the nested surd. Could a complex solution give a real final result?

[dan-florinchereches]

If √(a-√b)=√c-√d then
a=c+d
b=4cd

Then a=c+b/4c <=> 4c^2-4ac+b=0 so c = (4a+√(16a^2-16b))/8= (a+√(a^2-b))/2 and d =(a-√(a^2-b))/2

This makes the radical in the problem:

√(a-√(a^2-9))=1/√2( 1/√2(√(a+3)-√(a-3))

[yurenchu]

assuming |a| ≥ 3 :

a - √(a² - 9) =
= a - √((a+3)(a-3))
= a - √(a+3) * √(a-3)
= [ 2a - 2* √(a+3) * √(a-3) ]/2
= [ (a+3) + (a-3) - 2* √(a+3) * √(a-3) ]/2
... note: x² + y² - 2xy = (x-y)², here with x² = (a+3) and y² = (a-3) ...
= [ (√(a+3) - √(a-3))² ]/2
= [ (√(a+3) - √(a-3))/√2 ]²

and therefore,

√( a - √(a² - 9) ) =
= √( [ (√(a+3) - √(a-3))/√2 ]² )
= (√(a+3) - √(a-3)) / √2
= (√(a+3) - √(a-3)) * (√2)/2
= ( √(2a+6) - √(2a-6) ) / 2
= ½√(2a+6) - ½√(2a-6)