Question

The square root of \( 7 + 24i \) is:

• A. \( 4 - 3i \)
• B. \( 3 + 4i \)
• C. \( 3 - 4i \)
• D. \( 4 + 3i \)

Hint:

For square roots of complex numbers, convert the complex number to polar form and apply the square root formula for complex numbers. Then convert back to rectangular form.

Answer 1

The Correct Option is D

We are asked to find the square root of the complex number \( 7 + 24i \). To find \( \sqrt{7 + 24i} \), we use the fact that the square root of a complex number \( z = x + yi \) can be expressed as \( \sqrt{r}(\cos \theta + i \sin \theta) \), where \( r = \sqrt{x^2 + y^2} \) and \( \theta = \tan^{-1}(\frac{y}{x}) \). Here, \( 7 + 24i \) has \( x = 7 \) and \( y = 24 \). Step 1: Finding the magnitude The magnitude \( r \) is: \[ r = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25. \] Step 2: Finding the argument The argument \( \theta \) is: \[ \theta = \tan^{-1}\left( \frac{24}{7} \right) \approx 74.05^\circ. \] Step 3: Applying the square root formula Using the formula for the square root of a complex number, the square root of \( 7 + 24i \) is: \[ \sqrt{7 + 24i} = \sqrt{25} \left( \cos \frac{74.05^\circ}{2} + i \sin \frac{74.05^\circ}{2} \right). \] After calculating, we find that: \[ \sqrt{7 + 24i} = 4 + 3i. \] Thus, the square root of \( 7 + 24i \) is \( 4 + 3i \).

Answer 2

Problem: Find the square root of the complex number \[ 7 + 24i. \]

Step 1: Let the square root be \[ a + bi, \] where \( a \) and \( b \) are real numbers.

Step 2: Write the equation \[ (a + bi)^2 = 7 + 24i. \] Expanding the left side: \[ a^2 + 2ab i + b^2 i^2 = a^2 - b^2 + 2ab i. \] Equate real and imaginary parts: \[ a^2 - b^2 = 7, \] \[ 2ab = 24. \]

Step 3: Solve the system From the imaginary part: \[ 2ab = 24 \implies ab = 12. \] From the real part: \[ a^2 - b^2 = 7. \] Express \( b \) in terms of \( a \): \[ b = \frac{12}{a}. \] Substitute into the real part equation: \[ a^2 - \left(\frac{12}{a}\right)^2 = 7, \] \[ a^2 - \frac{144}{a^2} = 7. \] Multiply both sides by \( a^2 \): \[ a^4 - 144 = 7a^2. \] Rearranged: \[ a^4 - 7a^2 - 144 = 0. \] Let \( y = a^2 \): \[ y^2 - 7y - 144 = 0. \] Solve quadratic for \( y \): \[ y = \frac{7 \pm \sqrt{49 + 576}}{2} = \frac{7 \pm \sqrt{625}}{2} = \frac{7 \pm 25}{2}. \] Two solutions: \[ y = \frac{7 + 25}{2} = 16, \quad y = \frac{7 - 25}{2} = -9 \quad (\text{discard negative}). \] So, \[ a^2 = 16 \implies a = \pm 4. \] From \( ab = 12 \), if \( a = 4 \), then \[ b = \frac{12}{4} = 3. \]

Step 4: Final answer \[ \sqrt{7 + 24i} = 4 + 3i. \] (The other root is \( -4 - 3i \).)

Answer: \[ \boxed{4 + 3i}. \]