Square root is a one to one mapping

Algebra Level 5

C o n j u g a t e ( x ) = 1 + 1 − x Conjugate (x) = \sqrt {1 + \sqrt{1 - x}}

Which of the following is an answer to the above equation?

In the options, j = − 1 j = \sqrt{-1 } .

1.6180339887498948482045868343656 1.6180339887498948482045868343656 0.22298594482978 + j 0.41336379625112 1.12114689645950 + j 0.18434863333096 0.6180339887498948482045868343656 0.6180339887498948482045868343656 0 − 0.6180339887498948482045868343656 -0.6180339887498948482045868343656

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Lu Chee Ket by Lu Chee Ket , 47, Malaysia

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1 solution

Lu Chee Ket
Feb 24, 2015

1 Helpful 0 Interesting 0 Brilliant 0 Confused

Can you explain how you arrived at your answer?

Even if we square both sides repeatedly, we arrive at

( x 2 − 1 ) 2 = 1 − x ( x^2 -1 ) ^ 2 = 1 -x

which is equal to ( x ( x − 1 ) ( x 2 + x − 1 ) = 0 (x ( x-1) ( x^2 + x - 1 ) = 0 . The roots are thus x = 0 , 1 , 1 2 ( − 1 ± 5 ) x = 0, 1, \frac{1}{2} ( -1 \pm \sqrt{5} ) .

Note that we will still have to verify that these are indeed valid solutions, which they might not be. (e.g. x=0 is not a solution)

Calvin Lin Staff - 6 years, 3 months ago

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