Determine the square root of the complex number \(7 - 24i\):

Determine the square root of the complex number \(7 - 24i\):

[A] 3 + 4i

[B] 5 - 3i

[C] 4 - 3i

[D] 3 - 5i

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**Solution:**

Let's assume the square root of \(7 - 24i\) is \(a + bi\), where \(a\) and \(b\) are real numbers.

When squared:

\((a + bi)^2 = 7 - 24i\)

Expanding and equating the real and imaginary parts:

Real Part: \(a^2 - b^2 = 7\)

Imaginary Part: \(2ab = -24\) => \(ab = -12\)

From the second equation, using combinations like \(a = 3, b = -4\) or \(a = -4, b = 3\), the combination \(a = 3, b = -4\) satisfies the first equation \(a^2 - b^2 = 7\).

Thus, the square root of \(7 - 24i\) is:

\(3 - 4i\)

**Correct Answer: [A] 3 + 4i**

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This MCQ further explores the topic of finding square roots of complex numbers.