Let z and w be complex numbers such that |2z - w| = 25, |z + 2w| = 5, and |z + w| = 2. Find |z|.

Question

I need a little help with absolute values on complex numbers.

Answer 1

You asked for it:

2Z-W=(2Zx-Wx)i+(2Zy-Wy)j
|2Z-W|^2=4Zx^2-4ZxWx+Wx^2+4Zy-4ZyWy+Wy^2
=4|Z|^2+|W|^2 -4(ZxWx+ZyWy)= 625
and
|Z+2W|^2 =|Z|^2+4|W|^2 +4(ZxWx+ZyWy)=25
so
625+25 =5[ |Z|^2+|W|2| ]
or
130 = |Z|^2 + |W|^2
now that last one
|Z+W|^2= Zx^2+2ZxWx+Wx^2+Zy^2+2ZyWy+Wy^2] = 4
or
|Z|^2+|W|^2 +2(ZxWx+ZyWy)=4
or
ZxWx+ZyWy=-63
go back up
625 = 4|Z|^2+|W|^2+4*63
25 = |Z|^2+4|W|^2 -4*63
------------------------
or
2500 = 16|Z|^2 + 4 |W|^2 +16(63)
25 = |Z|^2 + 4 |W|^2 -4(63)
------------------------------
2475 = 15 |Z|^2 + 20(63)
all yours :)

Answer 2

at least one typo

2Z-W=(2Zx-Wx)i+(2Zy-Wy)j
|2Z-W|^2=4Zx^2-4ZxWx+Wx^2+4Zy-4ZyWy+Wy^2
=4|Z|^2+|W|^2 -4(ZxWx+ZyWy)= 625
and
|Z+2W|^2 =|Z|^2+4|W|^2 +4(ZxWx+ZyWy)=25
so
625+25 =5[ |Z|^2+|W|^2| ]
or
130 = |Z|^2 + |W|^2
now that last one
|Z+W|^2= Zx^2+2ZxWx+Wx^2+Zy^2+2ZyWy+Wy^2] = 4
or
|Z|^2+|W|^2 +2(ZxWx+ZyWy)=4
or
ZxWx+ZyWy=-63
go back up
625 = 4|Z|^2+|W|^2+4*63
25 = |Z|^2+4|W|^2 -4*63
------------------------
or
2500 = 16|Z|^2 + 4 |W|^2 +16(63)
25 = |Z|^2 + 4 |W|^2 -4(63)
------------------------------
2475 = 15 |Z|^2 + 20(63)

Answer 3

Questions that may come in exams about complex numbers

Answer 4

mathematics Foundation