Solve `|x^2-2x-8|=2x`

Answer

To solve the equation \( |x^2 - 2x - 8| = 2x \), we will consider two cases based on the definition of absolute value.

### Step 1: Identify the cases
The expression inside the absolute value can be either positive or negative. Thus, we have two cases to consider:

1. **Case 1:** \( x^2 - 2x - 8 \geq 0 \)
2. **Case 2:** \( x^2 - 2x - 8 < 0 \)

### Step 2: Solve Case 1
For **Case 1**, we have:
\[ x^2 - 2x - 8 = 2x \]
Rearranging gives:
\[ x^2 - 4x - 8 = 0 \]
Now, we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -4, c = -8 \):
\[ x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \]
\[ x = \frac{4 \pm \sqrt{16 + 32}}{2} \]
\[ x = \frac{4 \pm \sqrt{48}}{2} \]
\[ x = \frac{4 \pm 4\sqrt{3}}{2} \]
\[ x = 2 \pm 2\sqrt{3} \]
Thus, the solutions for Case 1 are:
\[ x = 2 + 2\sqrt{3} \quad \text{and} \quad x = 2 - 2\sqrt{3} \]

### Step 3: Check the conditions for Case 1
We need to check whether these solutions satisfy the condition \( x^2 - 2x - 8 \geq 0 \).

1. **For \( x = 2 + 2\sqrt{3} \)**:
\[ x^2 - 2x - 8 = (2 + 2\sqrt{3})^2 - 2(2 + 2\sqrt{3}) - 8 \]
After simplification, it can be shown that this value is positive.

2. **For \( x = 2 - 2\sqrt{3} \)**:
\[ x^2 - 2x - 8 = (2 - 2\sqrt{3})^2 - 2(2 - 2\sqrt{3}) - 8 \]
This value is negative, thus it does not satisfy the condition.

### Step 4: Solve Case 2
For **Case 2**, we have:
\[ -(x^2 - 2x - 8) = 2x \]
This simplifies to:
\[ -x^2 + 2x + 8 = 2x \]
\[ -x^2 + 8 = 0 \]
\[ x^2 = 8 \]
Thus, we find:
\[ x = \pm 2\sqrt{2} \]

### Step 5: Check the conditions for Case 2
We need to check whether these solutions satisfy the condition \( x^2 - 2x - 8 < 0 \).

1. **For \( x = 2\sqrt{2} \)**:
\[ x^2 - 2x - 8 = (2\sqrt{2})^2 - 2(2\sqrt{2}) - 8 \]
This value is negative, thus it satisfies the condition.

2. **For \( x = -2\sqrt{2} \)**:
\[ x^2 - 2x - 8 = (-2\sqrt{2})^2 - 2(-2\sqrt{2}) - 8 \]
This value is positive, thus it does not satisfy the condition.

### Final Solutions
The valid solutions to the equation \( |x^2 - 2x - 8| = 2x \) are:
\[ x = 2 + 2\sqrt{3} \quad \text{and} \quad x = 2\sqrt{2} \]