Mix of Z and |Z|

Algebra Level 3

The quantity $Z$ is a complex number which satisfies the following equation:

$\large{Z^2 + |Z|^2 - 3 + 5 j = 0}$

What is $|Z|$ ?

Details and Assumptions:
1) $j = \sqrt{-1}$
2) $|\cdot|$ denotes the magnitude of a complex number

2 solutions

Chris Lewis
Jul 22, 2019

Let $Z=X+iY$ with $X,Y$ real. Then $Z^2=X^2+2iXY-Y^2$ and $|Z|^2=X^2+Y^2$ .

Considering the real parts of the equation,

$X^2-Y^2+X^2+Y^2-3=0$

so $X^2=\frac32$ . Considering imaginary parts,

$2XY+5=0$

so $Y^2=\frac{25}{6}$ .

Hence $|Z|^2=X^2+Y^2=\frac{17}{3}$ and $|Z|=\boxed{\sqrt{\frac{17}{3}}}$ .

A Former Brilliant Member
Jul 21, 2019

Z= $√(\dfrac{3}{2})$ - $\dfrac{5}{√6}j$ . So $|Z|=$ $√(\dfrac{17}{3})=2.380476$

Is this the only solution?

Pi Han Goh - 1 year, 10 months ago

No. There is another solution also. But that does not change the answer.

A Former Brilliant Member - 1 year, 10 months ago

Can you prove this assertion?

Pi Han Goh - 1 year, 10 months ago

Take Z=a+bj, substitute in the given equation, compare the real and imaginary parts.

A Former Brilliant Member - 1 year, 10 months ago