Solve `|(x-1)/(3+2x-8x^(2))|+|1-x|=(x-1)^(2)/(|3+2x-8x^(2)|) +1`

Answer

To solve the equation

\[ \left|\frac{x-1}{3 + 2x - 8x^2}\right| + |1-x| = \frac{(x-1)^2}{|3 + 2x - 8x^2|} + 1, \]

we will break it down step by step.

### Step 1: Rewrite the Equation
Let:
- \( a = \frac{x-1}{3 + 2x - 8x^2} \)
- \( b = x - 1 \)

The equation then becomes:

\[ |a| + |1 - b| = \frac{b^2}{|3 + 2x - 8x^2|} + 1. \]

### Step 2: Analyze the Absolute Values
We know that:

\[ |1 - b| = |1 - (x - 1)| = |2 - x|. \]

Thus, the equation can be rewritten as:

\[ |a| + |2 - x| = \frac{b^2}{|3 + 2x - 8x^2|} + 1. \]

### Step 3: Set Up Cases for Absolute Values
We need to consider different cases based on the values of \( a \) and \( b \).

1. **Case 1**: \( a \geq 0 \) and \( b \geq 0 \)
2. **Case 2**: \( a \geq 0 \) and \( b < 0 \)
3. **Case 3**: \( a < 0 \) and \( b \geq 0 \)
4. **Case 4**: \( a < 0 \) and \( b < 0 \)

### Step 4: Solve Each Case
#### Case 1: \( a \geq 0 \) and \( b \geq 0 \)
This gives us:

\[ a + (2 - x) = \frac{b^2}{a} + 1. \]

Substituting \( a \) and \( b \):

\[ \frac{x-1}{3 + 2x - 8x^2} + (2 - x) = \frac{(x-1)^2}{\left|3 + 2x - 8x^2\right|} + 1. \]

#### Case 2: \( a \geq 0 \) and \( b < 0 \)
This gives us:

\[ a + (x - 2) = \frac{b^2}{a} + 1. \]

#### Case 3: \( a < 0 \) and \( b \geq 0 \)
This gives us:

\[ -a + (2 - x) = \frac{b^2}{-a} + 1. \]

#### Case 4: \( a < 0 \) and \( b < 0 \)
This gives us:

\[ -a + (x - 2) = \frac{b^2}{-a} + 1. \]

### Step 5: Solve the Quadratic Equations
After simplifying the equations from each case, we will end up with quadratic equations. For example:

1. From Case 1, we might derive:

\[ 8x^2 - x - 4 = 0. \]

2. From Case 2, we might derive:

\[ 8x^2 - 3x - 2 = 0. \]

### Step 6: Use the Quadratic Formula
For the equations derived, we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \]

For example, for \( 8x^2 - x - 4 = 0 \):

- \( a = 8, b = -1, c = -4 \)
- Discriminant: \( (-1)^2 - 4 \cdot 8 \cdot (-4) = 1 + 128 = 129 \)
- Solutions:

\[ x = \frac{1 \pm \sqrt{129}}{16}. \]

### Step 7: Collect All Solutions
From all cases, we will find the values of \( x \):

1. From Case 1: \( x = \frac{1 \pm \sqrt{129}}{16} \)
2. From Case 2: \( x = \frac{3 \pm \sqrt{73}}{16} \)
3. From Case 3 and Case 4 will yield additional values.

### Final Step: Verify Solutions
We need to check each solution against the original equation to ensure they satisfy it.