SOLUTION: Find all the solutions to the equation {{{2x^2-1=isqrt(3)}}} I missed a lecture and don't know the technique for finding the roots of equations like this which contain complex num

Algebra ->  Complex Numbers Imaginary Numbers Solvers and Lesson -> SOLUTION: Find all the solutions to the equation {{{2x^2-1=isqrt(3)}}} I missed a lecture and don't know the technique for finding the roots of equations like this which contain complex num      Log On


   

Question 981441: Find all the solutions to the equation 2x%5E2-1=isqrt%283%29
I missed a lecture and don't know the technique for finding the roots of equations like this which contain complex numbers. Can anyone briefly walk me through how to approach this?
Answer by josgarithmetic(39627) About Me  (Show Source):
You can put this solution on YOUR website!
One of the tutors made a lesson on this which you can read here:
http://www.algebra.com/algebra/homework/complex/How-to-take-a-square-root-of-a-complex-number.lesson

Here is another way you might be able to try, here not done to completion:

What you show is equivalent to x%5E2=1%2F2%2B%28sqrt%283%29%2F2%29i.
Assume you have some unknown x=a%2Bbi.

x%5E2=%28a%2Bbi%29%28a%2Bbi%29=1%2F2%2B%28sqrt%283%29%2F2%29i
a%5E2%2B2abi%2Bbi%5E2
a%5E2-b%5E2%2B2abi which must be equal to 1%2F2%2Bi%28sqrt%283%29%2F2%29.

This means two equations by corresponding parts,
highlight_green%28a%5E2-b%5E2=1%2F2%29 and highlight_green%282ab=sqrt%283%29%2F2%29.
Solving for a and b should be possible.

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A little algebra work using these two equations should bring you to
highlight_green%2816a%5E4-8a%5E2-3=0%29, a quadratic form equation in the variable, a. This will be complicated; solving first for a^2, and then for a.
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Formula for the general solution of a quadratic equation will give you
a%5E2=-1%2F4 or a%5E2=3%2F4.
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a=-i%2F2 or a=i%2F2 or a=-sqrt%283%29i%2F2 or a=sqrt%283%29i%2F2.
You could find the value for b corresponding to each of these.