Find the square root of -6+8i.

Answer

To find the square root of the complex number \(-6 + 8i\), we can follow these steps:

### Step 1: Assume the square root
Let \( z = a + bi \) be the square root of \(-6 + 8i\). Therefore, we have:
\[ z^2 = -6 + 8i \]

### Step 2: Expand \( z^2 \)
Expanding \( z^2 \):
\[ (a + bi)^2 = a^2 + 2abi + (bi)^2 = a^2 - b^2 + 2abi \]
This gives us:
\[ z^2 = a^2 - b^2 + 2abi \]

### Step 3: Set up equations
Now, we can equate the real and imaginary parts:
1. \( a^2 - b^2 = -6 \) (real part)
2. \( 2ab = 8 \) (imaginary part)

### Step 4: Solve for \( b \)
From the second equation, we can express \( b \) in terms of \( a \):
\[ b = \frac{8}{2a} = \frac{4}{a} \]

### Step 5: Substitute \( b \) into the first equation
Substituting \( b \) into the first equation:
\[ a^2 - \left(\frac{4}{a}\right)^2 = -6 \]
This simplifies to:
\[ a^2 - \frac{16}{a^2} = -6 \]

### Step 6: Multiply through by \( a^2 \)
To eliminate the fraction, multiply through by \( a^2 \):
\[ a^4 + 6a^2 - 16 = 0 \]

### Step 7: Let \( x = a^2 \)
Let \( x = a^2 \). Then we have:
\[ x^2 + 6x - 16 = 0 \]

### Step 8: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
\[ x = \frac{-6 \pm \sqrt{6^2 - 4 \cdot 1 \cdot (-16)}}{2 \cdot 1} = \frac{-6 \pm \sqrt{36 + 64}}{2} = \frac{-6 \pm \sqrt{100}}{2} \]
\[ x = \frac{-6 \pm 10}{2} \]
This gives us two solutions:
\[ x = 2 \quad \text{and} \quad x = -8 \]

### Step 9: Find \( a \)
Since \( x = a^2 \), we only take the positive solution:
\[ a^2 = 2 \implies a = \sqrt{2} \quad \text{or} \quad a = -\sqrt{2} \]

### Step 10: Find \( b \)
Using \( b = \frac{4}{a} \):
1. If \( a = \sqrt{2} \):
\[ b = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]
2. If \( a = -\sqrt{2} \):
\[ b = \frac{4}{-\sqrt{2}} = -2\sqrt{2} \]

### Step 11: Write the square roots
Thus, the two square roots of \(-6 + 8i\) are:
\[ \sqrt{2} + 2\sqrt{2}i \quad \text{and} \quad -\sqrt{2} - 2\sqrt{2}i \]

### Final Answer
The square roots of \(-6 + 8i\) are:
\[ \sqrt{2} + 2\sqrt{2}i \quad \text{and} \quad -\sqrt{2} - 2\sqrt{2}i \]