Complex Variables

Ariana

Hi,

I'd appreciate if someone could help me understand how to solve these problems. I've seen proofs and all being done but I don't really understand the logic behind it. They are probably really simple questions tho.

1) Is there a number of the form a + b√2 (where a, b ∈ Q[rational #s]) whose square is equal to 7? (if yes, give an example and if no, prove it).

2) For the equation X + 2X + 16 = 0.
a. Does it have a solution in Q[rational #s]?
b. Does it have a solution in R[real #s]?
c. Does it have a solution in C[complex #s]?
When it does have a solution, solve completely the equation. If it doesn’t, prove that it doesn’t.

Thanks for help in advance.

**Posted:** Thu Sep 18, 2008 5:54 pm

hqthao · New Member Offline Joined: 15 May 2009 Posts: 9

first, i want to tell you that all the equation with n degree will have n roots in complex field, but we have already proved that we just solved equation by square roots for the equation with 4,3,2,1 degree. in your equation, because it doesn't have roots in real field so will doesn't have in Q field (of course) Mr. Green

have fun

**Posted:** Fri May 15, 2009 4:28 am

grn_trtle

First of all, $\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}$ , so #2, being a polynomial, has 2 complex roots (Fundamental Theorem of Algebra). I'm assuming you meant $x^2+2x+16=0$ , because otherwise it's linear and the solution is obvious. This parabola has a vertex above the $x$ axis (completing the square can prove that it's positive for all $x\in\mathbb{R}$ ), so all its roots are non-real.

You can solve it with the quadratic formula.

**Posted:** Fri May 15, 2009 10:12 am

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t0rajir0u

**Posted:** Fri May 15, 2009 10:24 am

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