Taking the value of the square root with positive real part only, the value of `sqrt(7+24i)+sqrt(-7-24i)`, is

Options

A. `1+7i`
B. `-1-7i`
C. `7-i`
D. `-7+i`

Explanation

To solve the expression \( \sqrt{7 + 24i} + \sqrt{-7 - 24i} \), we will follow these steps:

### Step 1: Find \( \sqrt{7 + 24i} \)

Assume \( \sqrt{7 + 24i} = x + yi \), where \( x \) and \( y \) are real numbers.

Squaring both sides gives:
\[ 7 + 24i = (x + yi)^2 = x^2 - y^2 + 2xyi \]

### Step 2: Equate real and imaginary parts

From the equation \( 7 + 24i = x^2 - y^2 + 2xyi \), we can equate the real and imaginary parts:
1. \( x^2 - y^2 = 7 \) (1)
2. \( 2xy = 24 \) (2)

From equation (2), we can simplify to find \( xy \):
\[ xy = 12 \]

### Step 3: Substitute \( y \) in terms of \( x \)

From \( xy = 12 \), we can express \( y \) as:
\[ y = \frac{12}{x} \]

### Step 4: Substitute \( y \) into the first equation

Substituting \( y \) into equation (1):
\[ x^2 - \left(\frac{12}{x}\right)^2 = 7 \]
This simplifies to:
\[ x^2 - \frac{144}{x^2} = 7 \]

### Step 5: Multiply through by \( x^2 \)

To eliminate the fraction, multiply through by \( x^2 \):
\[ x^4 - 144 = 7x^2 \]
Rearranging gives:
\[ x^4 - 7x^2 - 144 = 0 \]

### Step 6: Let \( z = x^2 \)

Let \( z = x^2 \), then we have:
\[ z^2 - 7z - 144 = 0 \]

### Step 7: Solve the quadratic equation

Using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ z = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot (-144)}}{2 \cdot 1} \]
\[ z = \frac{7 \pm \sqrt{49 + 576}}{2} \]
\[ z = \frac{7 \pm \sqrt{625}}{2} \]
\[ z = \frac{7 \pm 25}{2} \]

Calculating the two possible values:
1. \( z = \frac{32}{2} = 16 \) (valid)
2. \( z = \frac{-18}{2} = -9 \) (not valid)

Thus, \( x^2 = 16 \) implies \( x = 4 \) (taking the positive root since we want the positive real part).

### Step 8: Find \( y \)

Now substituting \( x = 4 \) back into \( y = \frac{12}{x} \):
\[ y = \frac{12}{4} = 3 \]

Thus, we have:
\[ \sqrt{7 + 24i} = 4 + 3i \]

### Step 9: Find \( \sqrt{-7 - 24i} \)

Assume \( \sqrt{-7 - 24i} = a + bi \). Squaring gives:
\[ -7 - 24i = (a + bi)^2 = a^2 - b^2 + 2abi \]

Equating real and imaginary parts:
1. \( a^2 - b^2 = -7 \) (3)
2. \( 2ab = -24 \) (4)

From equation (4):
\[ ab = -12 \]

### Step 10: Substitute \( b \) in terms of \( a \)

From \( ab = -12 \):
\[ b = \frac{-12}{a} \]

### Step 11: Substitute \( b \) into the first equation

Substituting into equation (3):
\[ a^2 - \left(\frac{-12}{a}\right)^2 = -7 \]
This simplifies to:
\[ a^2 - \frac{144}{a^2} = -7 \]

### Step 12: Multiply through by \( a^2 \)

Multiply through by \( a^2 \):
\[ a^4 + 7a^2 - 144 = 0 \]

### Step 13: Let \( w = a^2 \)

Let \( w = a^2 \):
\[ w^2 + 7w - 144 = 0 \]

### Step 14: Solve the quadratic equation

Using the quadratic formula:
\[ w = \frac{-7 \pm \sqrt{7^2 + 4 \cdot 144}}{2} \]
\[ w = \frac{-7 \pm \sqrt{49 + 576}}{2} \]
\[ w = \frac{-7 \pm 25}{2} \]

Calculating the two possible values:
1. \( w = \frac{18}{2} = 9 \) (valid)
2. \( w = \frac{-32}{2} = -16 \) (not valid)

Thus, \( a^2 = 9 \) implies \( a = 3 \) (taking the positive root).

### Step 15: Find \( b \)

Now substituting \( a = 3 \) back into \( b = \frac{-12}{a} \):
\[ b = \frac{-12}{3} = -4 \]

Thus, we have:
\[ \sqrt{-7 - 24i} = 3 - 4i \]

### Step 16: Combine the results

Now we can combine the results:
\[ \sqrt{7 + 24i} + \sqrt{-7 - 24i} = (4 + 3i) + (3 - 4i) \]
\[ = 4 + 3 + (3i - 4i) = 7 - i \]

### Final Answer

The value of \( \sqrt{7 + 24i} + \sqrt{-7 - 24i} \) is:
\[ \boxed{7 - i} \]