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I have the following to multiply ;

$$(z-p-qi+\sqrt{t+ui})(z-p+qi+\sqrt{t-ui})$$

Now, I think that the product must not have any complex numbers...

But here is what I get

$$z^2-2zp+p^2+q^1+\sqrt{t^2+u^2}+(\sqrt{t-ui}+\sqrt{t+ui})(z-p+qi)$$

I'm pretty sure that there is some step that I'm missing... Can anybody point me out my error ? Thanks

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  • $\begingroup$ why are you assuming that the answer will not have any complex numbers? In fact the two numbers you are multiplying need not be complex conjugates (because of the presence of $z$). $\endgroup$ – Anurag A Apr 21 '15 at 16:35
  • $\begingroup$ For no particular reason, I just thought that they needed to go... $\endgroup$ – user108343 Apr 21 '15 at 16:36
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I think you made a mistake about sign. See the red sign at the last. $$(z-p-qi+\sqrt{t+ui})(z-p+qi+\sqrt{t-ui})$$ $$=z^2+z(-p+qi)+z\sqrt{t-ui}+z(-p-qi)+(-p-qi)(-p+qi)+(-p-qi)\sqrt{t-ui}+z\sqrt{t+ui}+(-p+qi)\sqrt{t+ui}+\sqrt{(t+ui)(t-ui)}$$ $$=z^2+z(-p+qi-p-qi)+z(\sqrt{t-ui}+\sqrt{t+ui})+p^2+q^2+(-p-qi)\sqrt{t-ui}+(-p+qi)\sqrt{t+ui}+\sqrt{t^2+u^2}$$ $$=z^2-2pz+z(\sqrt{t-ui}+\sqrt{t+ui})+p^2+q^2+(-p-qi)\sqrt{t-ui}+(-p+qi)\sqrt{t+ui}+\sqrt{t^2+u^2}$$ $$=z^2-2pz+p^2+q^2+\sqrt{t^2+u^2}+(z-p)(\sqrt{t-ui}+\sqrt{t+ui})+qi(\sqrt{t+ui}\color{red}{-}\sqrt{t-ui})$$

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  • $\begingroup$ Mmmmh...My two factors are written in the correct way... But, as I see, we can't get rid of the complex parts, can't we ?? $\endgroup$ – user108343 Apr 21 '15 at 17:06
  • $\begingroup$ @Astroman: I think we cannot. Why do you think that it must not have any complex numbers? Because of some answers in a book? $\endgroup$ – mathlove Apr 21 '15 at 17:10
  • $\begingroup$ Yes, exactly. I guess it's some error $\endgroup$ – user108343 Apr 21 '15 at 17:10
  • $\begingroup$ @Astroman: Can you write the answer? $\endgroup$ – mathlove Apr 21 '15 at 17:11
  • $\begingroup$ Here : imgur.com/LTbx3hS The idea is that we want to express it in a quadratic form.. $\endgroup$ – user108343 Apr 21 '15 at 17:17

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