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Tutor question #3 Simplify: \[\sqrt{5-\sqrt{21}}\] (conditions: pen, paper, no calculator, done in 2 minutes) *

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what if x = \(\sqrt{5-\sqrt{21}}\) then \(x^2 = 5 - \sqrt{21}\) \[x^2 - 5 = -\sqrt{21}\] \[(x^2 - 5)^2 = 21\] oh goodness quadratic o.O
actually it is a 4th degree equation
\[(x^2-5)^2=21\]\[x^4-10x^2+25=21\]\[x^4-10x^2+4=0\]\[x^4-10x^2+4=0\]\[x^2=\frac{10\pm \sqrt{(-10)^6-4(4)}}{2}\]\[x^2=5\pm \sqrt{21}\]Sorry... are you sure that it is 'simplified'?

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Other answers:

Oh.that's power 2, typo
apparently it is also \[\sqrt{\frac{7}{2}}-\sqrt{\frac{3}{2}}\] i remember seeing something like this before but i am not sure i remember how to go from one to the other
I need to know how to work that out.....
cant seem to get it, i think it was a trick
............................ It... was ... a ... question ... asked ... when ... my friend applied for a summer job ...
really? doing what??
tutor - teaching high school students
hmm i wonder what answer they wanted
Same here...
When I was doing some exercises few days ago, I saw similar questions, but clearer, like this: express \(\sqrt{28-2\sqrt{147}}\) in the form of \(\sqrt{x}-\sqrt{y}\). I can still handle this. But that one, I failed :(
multiplying by the conjugate give \(\frac{2}{\sqrt{5+\sqrt{21}}}\) think
any examples anywhere?
You want to show \[\sqrt{5-\sqrt{21}} \text{ equals } \frac{\sqrt{7}-\sqrt{3}}{2} ?\] was just wondering if you have an example for writing that one thing in that other form
oops sqrt(2) on bottom
how did you get that identity?
From experience...
lol great experience I want to prove that identity lol
First, \((\sqrt{a} - \sqrt{b})^2\) = a + b -2\sqrt{ab} For a>b \[\sqrt{5-\sqrt{21}} = \sqrt{5-2\sqrt{\frac{21}{4}}}\] Now, a+b = 5 => a=5-b ab = 21/4 (5-b)b = 21/4 -4b^2 + 20b - 21 =0 b=1.5 or b =3.5 (rejected) a = 5 - 1.5 = 3.5 So, it is \(\sqrt{\frac{7}{2}}-\sqrt{\frac{3}{2}}\) Does that make sense?
ok i see that identity :)
that one is easy to prove
maybe because it is an actual identity, right? :p
Perfect square is perfect :)
Does that make sense? Apart from the latex fail...
very interesting i wouldn't have thought of that
I'm going to post the link to satellite73's post then. He doesn't even come to check..
Great work @Callisto :)
And thank you for all your time!!!!

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