What is the number of distinct solutions of the equation z2 + |z| = 0(where z is a complex number)?a)Oneb)Twoc)Threed)FiveCorrect answer is option 'C'. Can you explain this answer?

Question

Answer

Distinct Solutions of the Equation z^2 + |z| = 0

Understanding the Equation:
The given equation is z^2 + |z| = 0, where z is a complex number.

Definition of Absolute Value:
The absolute value of a complex number z = a + bi is |z| = sqrt(a^2 + b^2).

Expressing z in Terms of Its Components:
Let z = x + yi, where x and y are real numbers.

Substitute z into the Equation:
(x + yi)^2 + sqrt(x^2 + y^2) = 0
Expanding and simplifying the equation gives:
x^2 - y^2 + 2xyi + sqrt(x^2 + y^2) = 0

Separating Real and Imaginary Parts:
Setting the real and imaginary parts of the equation to zero, we get:
Real Part: x^2 - y^2 + sqrt(x^2 + y^2) = 0
Imaginary Part: 2xy = 0

Finding Solutions:
From the imaginary part, we have x = 0 or y = 0.
Case 1: x = 0
Substitute x = 0 into the real part:
-y^2 + y = 0
y(y - 1) = 0
y = 0 or y = 1
Case 2: y = 0
Substitute y = 0 into the real part:
sqrt(x^2) = 0
x = 0

Combining Solutions:
Combining the solutions from both cases, we have the distinct solutions:
z = 0, z = i, z = -i
Therefore, the number of distinct solutions of the equation z^2 + |z| = 0 is three (z = 0, z = i, z = -i).

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