# Another Complex Number Question

• Jul 30th 2009, 01:15 AM
Stroodle
Another Complex Number Question
If $\displaystyle z=x+yi$ determine the values of $\displaystyle x$ and $\displaystyle y$ such that $\displaystyle z=\sqrt{3+4i}$
• Jul 30th 2009, 01:40 AM
red_dog
$\displaystyle 3+4i=4+4i-1=4+4i+i^2=(2+i)^2$

Then $\displaystyle z_1=2+i, \ z_2=-2-i$
• Jul 30th 2009, 01:48 AM
Stroodle
• Jul 30th 2009, 03:06 AM
mr fantastic
Quote:

Originally Posted by Stroodle
If $\displaystyle z=x+yi$ determine the values of $\displaystyle x$ and $\displaystyle y$ such that $\displaystyle z=\sqrt{3+4i}$

Another approach:

$\displaystyle z^2 = 3 + 4i$

$\displaystyle \Rightarrow (x + iy)^2 = 3 + 4i$

$\displaystyle \Rightarrow (x^2 - y^2) + 2xy i = 3 + 4i$.

Therefore:

$\displaystyle x^2 - y^2 = 3$ .... (1)

$\displaystyle 2xy = 4 \Rightarrow xy = 2 \Rightarrow y = \frac{2}{x}$ .... (2)

Substitute (2) into (1): $\displaystyle x^2 - \frac{4}{x^2} = 3 \Rightarrow x^4 - 3x^2 - 4 = 0 \Rightarrow (x^2 - 4)(x^2 + 1) = 0$.

But since x is real, $\displaystyle x = \pm 2$.

x = 2: y = 1.
x = -2: y = -1.
• Jul 30th 2009, 03:46 AM
Stroodle
Awesome. Thanks for that.
This method will be much easier for me to reproduce for similar problems.