Results 1 to 2 of 2

Thread: Another complex number

  1. #1
    MHF Contributor
    Joined
    Sep 2008
    From
    West Malaysia
    Posts
    1,261
    Thanks
    1

    Another complex number

    If $\displaystyle z=x+yi$ and $\displaystyle z^2=a+bi$ , x,y,a,b are all real numbers . Prove that $\displaystyle 2x^2=\sqrt{(a^2+b^2)}+a$
    By solving the equation $\displaystyle z^4+6z^2+25=0$ for $\displaystyle z^2$ or otherwise , express each root of the equation in the form of $\displaystyle x+yi$
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Nov 2008
    From
    France
    Posts
    1,458
    Quote Originally Posted by mathaddict View Post
    If $\displaystyle z=x+yi$ and $\displaystyle z^2=a+bi$ , x,y,a,b are all real numbers . Prove that $\displaystyle 2x^2=\sqrt{(a^2+b^2)}+a$
    By solving the equation $\displaystyle z^4+6z^2+25=0$ for $\displaystyle z^2$ or otherwise , express each root of the equation in the form of $\displaystyle x+yi$
    Hi

    If $\displaystyle z=x+yi$ then
    $\displaystyle z^2=x^2-y^2+2xyi$

    Then
    (i) $\displaystyle x^2-y^2=a$
    (ii) $\displaystyle 2xy=b$ then $\displaystyle y=\frac{b}{2x}$

    Substituting y in (i)
    $\displaystyle x^2 - \frac{b^2}{(2x)^2}=a$

    $\displaystyle (2x^2)^2 - 2a (2x^2) - b^2=0$

    Reduced discriminant is $\displaystyle a^2 + b^2$

    The only positive solution is therefore $\displaystyle 2x^2=\sqrt{a^2+b^2}+a$

    Equation $\displaystyle z^4+6z^2+25=0$ for $\displaystyle z^2$
    gives $\displaystyle z^2 = -3 + 4i$ or $\displaystyle z^2 = -3 - 4i$

    It is just an application with a=-3, b=4 and then a=-3 and b=-4
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: Oct 2nd 2010, 01:54 PM
  2. Replies: 3
    Last Post: Sep 13th 2010, 11:13 AM
  3. Complex number
    Posted in the Algebra Forum
    Replies: 1
    Last Post: Oct 16th 2009, 06:59 AM
  4. complex number
    Posted in the Algebra Forum
    Replies: 3
    Last Post: Oct 29th 2008, 06:00 PM
  5. [SOLVED] Complex D Name F D Number
    Posted in the Trigonometry Forum
    Replies: 3
    Last Post: Jan 14th 2007, 08:51 AM

Search Tags


/mathhelpforum @mathhelpforum