Stuck on imaginary number simultaneous equation

[Flyingpooman]

Solve this simultaneous equation:

z^2 - w^2 = -2+16i
z + iw = 4+4i

please help I tried to do this for so long and am struggling. I tried substituting z as well as w from the second equation into the first one but I just cannot end up with the values of z and w which are in the form of a+bi.

 

[Deleted member 4993]

Solve this simultaneous equation:

z^2 - w^2 = -2+16i
z + iw = 4+4i

please help I tried to do this for so long and am struggling. I tried substituting z as well as w from the second equation into the first one but I just cannot end up with the values of z and w which are in the form of a+bi.

[z + iw]² = [4+4i]²

continue.......

 

[pka]

(4i)^2 = 16 : you knew that?

Actually \(\displaystyle (4\bf{i})^2=-16\)

 

[lookagain]

Multiply [2] by 4, and rearrange:
z^2 - w^2 + 2 = 16i [1]
4z + 4iw - 16 = 16i [2]

Subtract to get:
z^2 - 4z - w^2 - 4iw + 18 = 0

You now have a quadratic...carry on...

Denis, but you have one quadratic equation in two variables.

How can z and w be isolated?

 

[pka]

Solve this simultaneous equation:
z^2 - w^2 = -2+16i
z + iw = 4+4i

Simple substitution is the most efficient way to solve this.

See this calculation. \(\displaystyle (z,w)=(2+3i,~1-2i)\) AND \(\displaystyle (z,w)=(2+i,~3-2i)\)

 

[Ishuda]

Solve this simultaneous equation:

z^2 - w^2 = -2+16i
z + iw = 4+4i

please help I tried to do this for so long and am struggling. I tried substituting z as well as w from the second equation into the first one but I just cannot end up with the values of z and w which are in the form of a+bi.

Assuming we are looking for values of z and w and that i is the 'positive' square root of -1, there is no solution IF both z and w are real as the equations are written. Thus either (or both) z and w must be complex numbers. So letting
z = z_r + i z_i
and
w = w_r + i w_i
where z_r, z_i, w_r, and w_i are real valued. The second equation gives
z = a + w = 4 + w_r + i (4 + w_i)
where a = 4 + 4 i. Equating real and imaginary parts, the first equation gives
(a) z_r²-z_i²-w_r²+w_i² = -2
(b) z_r z_i + w_r w_i = 8
See if you can go from there.

Hint: Start with (a) to get a linear equation in w_r and w_i