Find the square roots of the following:` 15-8i`

Explanation

To find the square roots of the complex number \( 15 - 8i \), we can express it in the form \( z = a + bi \), where \( z^2 = 15 - 8i \).

### Step 1: Assume the square root
Let \( z = x + yi \), where \( x \) and \( y \) are real numbers. Then, we have:
\[ z^2 = (x + yi)^2 = x^2 + 2xyi - y^2 \]
This can be rewritten as:
\[ z^2 = (x^2 - y^2) + (2xy)i \]

### Step 2: Set up equations
We want \( z^2 = 15 - 8i \). Therefore, we can equate the real and imaginary parts:
1. \( x^2 - y^2 = 15 \) (Equation 1)
2. \( 2xy = -8 \) (Equation 2)

### Step 3: Solve for \( y \)
From Equation 2, we can express \( y \) in terms of \( x \):
\[ y = \frac{-8}{2x} = \frac{-4}{x} \]

### Step 4: Substitute \( y \) into Equation 1
Substituting \( y \) into Equation 1 gives:
\[ x^2 - \left(\frac{-4}{x}\right)^2 = 15 \]
This simplifies to:
\[ x^2 - \frac{16}{x^2} = 15 \]

### Step 5: Multiply through by \( x^2 \)
To eliminate the fraction, multiply the entire equation by \( x^2 \):
\[ x^4 - 16 = 15x^2 \]
Rearranging gives:
\[ x^4 - 15x^2 - 16 = 0 \]

### Step 6: Let \( u = x^2 \)
Let \( u = x^2 \). Then we have a quadratic equation:
\[ u^2 - 15u - 16 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( u = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ u = \frac{15 \pm \sqrt{(-15)^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1} \]
\[ u = \frac{15 \pm \sqrt{225 + 64}}{2} \]
\[ u = \frac{15 \pm \sqrt{289}}{2} \]
\[ u = \frac{15 \pm 17}{2} \]
Calculating the two possible values for \( u \):
1. \( u = \frac{32}{2} = 16 \)
2. \( u = \frac{-2}{2} = -1 \) (not valid since \( u = x^2 \) must be non-negative)

So, \( x^2 = 16 \), hence \( x = 4 \) or \( x = -4 \).

### Step 8: Find \( y \)
Using \( y = \frac{-4}{x} \):
- If \( x = 4 \), then \( y = \frac{-4}{4} = -1 \).
- If \( x = -4 \), then \( y = \frac{-4}{-4} = 1 \).

### Step 9: Write the square roots
Thus, the square roots of \( 15 - 8i \) are:
\[ z_1 = 4 - i \quad \text{and} \quad z_2 = -4 + i \]

### Final Answer:
The square roots of \( 15 - 8i \) are \( 4 - i \) and \( -4 + i \).

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