- #1

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## Main Question or Discussion Point

I got this out of

The author shows that if a and b are some yet to be determined real numbers where:

[tex]\sqrt[3]{2+\sqrt{-121}} = a+b\sqrt{-1}[/tex]

[tex]\sqrt[3]{2-\sqrt{-121}} = a-b\sqrt{-1}[/tex]

Then he takes the first equation and cubes both sides, does a bunch of Algebra and gets:

[tex] 2+\sqrt{-121} = a(a^2-3b^2)+b(3a^2-b^2)\sqrt{-1}[/tex]

And says if this is equal to the complex number, [tex] 2+\sqrt{-121}[/tex] then the real and imaginary parts must be separately equal. Then he splits terms into:

[tex] a(a^2-3b^2) = 2 [/tex]

[tex]b(3a^2-b^2)\sqrt{-1}=11[/tex]

To find that a = 2 and b = 1, then says "With these results Bombelli showed that the Cardan solution is 4 and this is correct."

The very last part is where I don't understand how all that complex stuff arrives back at 4. Even though, through simple Algebra with the very first equation with have real solutions.. Any help would be awesome. Thanks.

*An Imaginary Tale: The Story of Sqrt(-1)*. In section 1.5 of the book, the author explains that Bombelli took x^{3}= 15x + 4 and found the real solutions: 4, -2±sqrt(3). But if you plug the equation into the Cardan forumla you get imaginaries. http://en.wikipedia.org/wiki/Cardan_formula#Cardano.27s_methodThe author shows that if a and b are some yet to be determined real numbers where:

[tex]\sqrt[3]{2+\sqrt{-121}} = a+b\sqrt{-1}[/tex]

[tex]\sqrt[3]{2-\sqrt{-121}} = a-b\sqrt{-1}[/tex]

Then he takes the first equation and cubes both sides, does a bunch of Algebra and gets:

[tex] 2+\sqrt{-121} = a(a^2-3b^2)+b(3a^2-b^2)\sqrt{-1}[/tex]

And says if this is equal to the complex number, [tex] 2+\sqrt{-121}[/tex] then the real and imaginary parts must be separately equal. Then he splits terms into:

[tex] a(a^2-3b^2) = 2 [/tex]

[tex]b(3a^2-b^2)\sqrt{-1}=11[/tex]

To find that a = 2 and b = 1, then says "With these results Bombelli showed that the Cardan solution is 4 and this is correct."

The very last part is where I don't understand how all that complex stuff arrives back at 4. Even though, through simple Algebra with the very first equation with have real solutions.. Any help would be awesome. Thanks.