Mark O. answered • 06/03/18
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Hi Yomna,
Are you meaning to ask:
If a2 + 1/a2=4 what is a4 + 1/a4, where a is a real number?
Mark O.
Hi Yomna,
Here is the solution. Please comment that you received it. Thanks. -Mark
a2 + 1/a2 = 4
Multiply through the equation by a2, and we get
a4 + 1 = 4a2
We can then rearrange this equation and write
a4 - 4a2 + 1 = 0
Let's change variables. Let b = a2. Then the equation becomes
b2 - 4b + 1 = 0
We can solve this equation using the quadratic formula and get
b = 2 +/- √3
So, we really have
a2 = 2 +/- √3
So, there are really two solutions,
a2 = 2 + √3; a2 = 2 - √3
If all works out, we should get the same result when we write a4 + 1/a4.
Let's first consider the solution a2 = 2 + √3.
a4 = a2•a2 = (2 + √3)(2 + √3) = 7 + 4√3.
a4 + 1/a4 = 7 + 4√3 + 1/(7 + 4√3)
a4 + 1/a4 = [(7 + 4√3)(7 + 4√3) + 1]/(7 + 4√3) getting a common denominator
a4 + 1/a4 = [49 + 56√3 + 48 + 1]/(7 + 4√3)
a4 + 1/a4 = (98 + 56√3)/(7 + 4√3)
a4 + 1/a4 = 14(7 + 4√3)/(7 + 4√3)
a4 + 1/a4 = 14 with the cancelation of (7 + 4√3).
If you plug in the solution a2 = 2 + √3 into a4 + 1/a4 and work through the same steps as above, you will also find the same result a4 + 1/a4 = 14.
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06/03/18
Mark O.
Regarding my last sentence, I meant to say:
If you plug in the solution a2 = 2 - √3 into a4 + 1/a4 and work through the same steps as above, you will also find the same result a4 + 1/a4 = 14.
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06/03/18
Yomna A.
06/03/18