Question

The number of solutions for the equation ${{\text{z}}^2} + |{\text{z| = 0}}$ is
$
  {\text{A}}{\text{. 1}} \\
  {\text{B}}{\text{. 2}} \\
  {\text{C}}{\text{. 3}} \\
  {\text{D}}{\text{. 4}} \\
$

Answer 1

Hint: We write the given complex number in terms of x and y and substitute in the equation. Comparing the real and imaginary components, we determine x and y. That gives us the values of z.

Step-by-step answer:
Here z is a complex number and it is of the form
z = x + iy --- (i = imaginary number, i = $\sqrt {\left( { - 1} \right)} $)

Now, ${{\text{z}}^2} = {\left( {{\text{x + iy}}} \right)^2}$
                 = ${{\text{x}}^2} + {\left( {{\text{iy}}} \right)^2} + {\text{2xiy}}$ -- (${{\text{i}}^2} = - 1$)
                 = ${{\text{x}}^2} - {{\text{y}}^2} + {\text{2xiy}}$


|z| = $\sqrt {{{\text{x}}^2} + {{\text{y}}^2}} $
Mod z, denoted by |z| is the absolute/scalar value of z and is given by the above.

Therefore, ${{\text{z}}^2} + |{\text{z| = 0}}$
$ \Rightarrow {{\text{x}}^2} - {{\text{y}}^2} + {\text{2xiy + }}\sqrt {{{\text{x}}^2} + {{\text{y}}^2}} = 0$
(0 can be written as 0 + i0)
$ \Rightarrow {{\text{x}}^2} - {{\text{y}}^2} + {\text{2xiy + }}\sqrt {{{\text{x}}^2} + {{\text{y}}^2}} = 0 + {\text{i0}}$

Comparing real components and imaginary components on both sides, we get

Real components: Imaginary components:

${{\text{x}}^2} - {{\text{y}}^2}{\text{ + }}\sqrt {{{\text{x}}^2} + {{\text{y}}^2}} = 0$ 2ixy = i0
                                    ⟹xy = 0
                                    ⟹x = 0 or y = 0

Case I:

When y = 0

Real part,
⟹${{\text{x}}^2}{\text{ + }}\sqrt {{{\text{x}}^2}} = 0$
⟹${{\text{x}}^2}{\text{ + |x|}} = 0$
⟹x = 0
If x = 0 and y = 0 ⟹ z = x + iy = 0

Case 2:

When x = 0

Real part,
⟹${\text{ - }}{{\text{y}}^2} + \sqrt {{{\text{y}}^2}} = 0$
⟹${\text{ - }}{{\text{y}}^2} + |{\text{y|}} = 0$
⟹|y| (|y| - 1) = 0
⟹y = 0, +1, -1

Here y cannot be 0 since that case already exists.
Hence, y = +1,-1

Thus we have, x = 0, y = +1 and x = 0, y = -1, therefore z = 0 + 1(i) = +i and z = 0 -1(i) = -i

The solutions of the equation ${{\text{z}}^2} + |{\text{z| = 0}}$ are 0, +i, -i.

Hence there are 3 solutions. Option C is the correct answer.

Note: The key in solving such types of problems is to write the complex number in terms of real number x terms and imaginary number y terms. Correctly comparing the real and imaginary components in the equation is a crucial step. We ignore the case y = 0 in step 2 as it is already covered in step 1.
And i =$\sqrt {\left( { - 1} \right)} $.