Solve `|x^(2)-3x-4|=9-|x^(2)-1|`

Answer

To solve the equation \( |x^2 - 3x - 4| = 9 - |x^2 - 1| \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ |x^2 - 3x - 4| = 9 - |x^2 - 1| \]

### Step 2: Factor the expressions inside the absolute values
The expression \( x^2 - 3x - 4 \) can be factored as:
\[ x^2 - 3x - 4 = (x - 4)(x + 1) \]
And the expression \( x^2 - 1 \) can be factored as:
\[ x^2 - 1 = (x - 1)(x + 1) \]

### Step 3: Set up the equation with the factored forms
Substituting the factored forms into the equation gives:
\[ |(x - 4)(x + 1)| = 9 - |(x - 1)(x + 1)| \]

### Step 4: Rearranging the equation
Rearranging the equation, we have:
\[ |(x - 4)(x + 1)| + |(x - 1)(x + 1)| = 9 \]

### Step 5: Consider cases based on the absolute values
We need to consider different cases based on the values of \(x\) that affect the signs of the expressions inside the absolute values.

#### Case 1: \(x \geq 4\)
In this case, both expressions are positive:
\[ (x - 4)(x + 1) + (x - 1)(x + 1) = 9 \]
Simplifying gives:
\[ (x^2 - 3x - 4) + (x^2 + 1) = 9 \]
\[ 2x^2 - 3x - 3 = 9 \]
\[ 2x^2 - 3x - 12 = 0 \]
Using the quadratic formula:
\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 2 \cdot (-12)}}{2 \cdot 2} \]
\[ x = \frac{3 \pm \sqrt{9 + 96}}{4} = \frac{3 \pm 11}{4} \]
This gives:
\[ x = 3.5 \quad \text{and} \quad x = -2 \]
Only \(x = 3.5\) is valid in this case.

#### Case 2: \(1 \leq x < 4\)
In this case, we have:
\[ -(x - 4)(x + 1) + (x - 1)(x + 1) = 9 \]
Simplifying gives:
\[ -(x^2 - 3x - 4) + (x^2 - 1) = 9 \]
\[ 3x + 3 - 1 = 9 \]
\[ 3x + 2 = 9 \quad \Rightarrow \quad 3x = 7 \quad \Rightarrow \quad x = \frac{7}{3} \]
This value is valid in this range.

#### Case 3: \(x < 1\)
Here, both expressions are negative:
\[ -(x - 4)(x + 1) - (x - 1)(x + 1) = 9 \]
Simplifying gives:
\[ -(x^2 - 3x - 4) - (x^2 - 1) = 9 \]
\[ -2x^2 + 3x + 5 = 9 \]
\[ -2x^2 + 3x - 4 = 0 \]
Using the quadratic formula:
\[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot (-2) \cdot (-4)}}{2 \cdot (-2)} \]
This gives:
\[ x = \frac{-3 \pm \sqrt{9 - 32}}{-4} \]
This results in no real solutions.

### Final Solutions
The valid solutions from the cases are:
\[ x = 3.5, \quad x = \frac{7}{3} \]

### Summary of Solutions
The solutions to the equation \( |x^2 - 3x - 4| = 9 - |x^2 - 1| \) are:
\[ x = 3.5, \quad x = \frac{7}{3} \]