Question

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Answer 1

Answer:

What is the solution of the equation z^2 + |z| = 0, where z is a complex number?

SBI Gold Loan!

(An elementary approach)

Let z=x+iy, where x,y∈R & i=−1−−−√.

Then z2+|z|=0

⟹(x+iy)2+x2+y2−−−−−−√=0

⟹(x2−y2)+x2+y2−−−−−−√+i⋅2xy=0

∴x2−y2+x2+y2−−−−−−√=0...(1)

&2xy=0

⟹xy=0

That is, either x=0 or y=0.

Case I––––––

When y=0.

(1)⟹x2+x2−−√=0

⟹x2+|x|=0

⟹|x|2+|x|=0

⟹x=0

We thus have: x=y=0.

∴z=0.

Case II–––––––

When x=0.

(1)⟹−y2+y2−−√=0

⟹−y2+|y|=0

⟹−|y|2+|y|=0

⟹|y|(|y|−1)=0

⟹|y|=0,1

⟹y=0,±1

But in this case, y≠0.

So, y=1,−1.

Thus we get:

x=0,y=1 and x=0,y=−1.

∴z=i,−i.

Hence z=0,i & −i are the required solutions of z2+|z|=0.

Answer 2

Answer:

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