Find a squar root for the complex number `7+24i`

Answer

To find a square root of the complex number \( 7 + 24i \), we can follow these steps:

### Step 1: Assume the Square Root
Let us assume that the square root of \( 7 + 24i \) can be expressed in the form:
\[ x + yi \]
where \( x \) and \( y \) are real numbers.

### Step 2: Square Both Sides
Squaring both sides gives us:
\[ (x + yi)^2 = 7 + 24i \]
Expanding the left-hand side:
\[ x^2 + 2xyi - y^2 = 7 + 24i \]
This can be separated into real and imaginary parts:
\[ x^2 - y^2 + 2xyi = 7 + 24i \]

### Step 3: Set Up Equations
From the above equation, we can equate the real and imaginary parts:
1. \( x^2 - y^2 = 7 \) (Equation 1)
2. \( 2xy = 24 \) (Equation 2)

### Step 4: Solve for \( y \)
From Equation 2, we can express \( y \) in terms of \( x \):
\[ y = \frac{24}{2x} = \frac{12}{x} \]

### Step 5: Substitute \( y \) in Equation 1
Substituting \( y \) into Equation 1:
\[ x^2 - \left(\frac{12}{x}\right)^2 = 7 \]
This simplifies to:
\[ x^2 - \frac{144}{x^2} = 7 \]
Multiplying through by \( x^2 \) to eliminate the fraction:
\[ x^4 - 7x^2 - 144 = 0 \]

### Step 6: Let \( z = x^2 \)
Let \( z = x^2 \). Then, we have a quadratic equation:
\[ z^2 - 7z - 144 = 0 \]

### Step 7: Solve the Quadratic Equation
Using the quadratic formula \( z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ z = \frac{7 \pm \sqrt{(-7)^2 - 4 \cdot 1 \cdot (-144)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ z = \frac{7 \pm \sqrt{49 + 576}}{2} \]
\[ z = \frac{7 \pm \sqrt{625}}{2} \]
\[ z = \frac{7 \pm 25}{2} \]
Thus, we have two possible values for \( z \):
1. \( z = \frac{32}{2} = 16 \)
2. \( z = \frac{-18}{2} \) (not valid since \( z = x^2 \) must be non-negative)

So, \( x^2 = 16 \) implies:
\[ x = \pm 4 \]

### Step 8: Find \( y \)
Substituting \( x = 4 \) into \( y = \frac{12}{x} \):
\[ y = \frac{12}{4} = 3 \]
Substituting \( x = -4 \):
\[ y = \frac{12}{-4} = -3 \]

### Step 9: Write the Square Roots
Thus, the square roots of \( 7 + 24i \) are:
\[ 4 + 3i \quad \text{and} \quad -4 - 3i \]

### Final Answer
The square roots of the complex number \( 7 + 24i \) are:
\[ \boxed{4 + 3i \quad \text{and} \quad -4 - 3i} \]

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