The set of values of 'c' for which the equation `x^(2)-4x-c-sqrt(8x^(2)-32x-8c)=0` has exactly two distinct real solutions, is (a,b) then find the value of (b-a).

Answer

To solve the equation \( x^2 - 4x - c - \sqrt{8x^2 - 32x - 8c} = 0 \) for the values of \( c \) that yield exactly two distinct real solutions, we will follow these steps:

### Step 1: Rearranging the Equation
First, we can rearrange the equation to isolate the square root:
\[ \sqrt{8x^2 - 32x - 8c} = x^2 - 4x - c \]

### Step 2: Squaring Both Sides
Next, we square both sides to eliminate the square root:
\[ 8x^2 - 32x - 8c = (x^2 - 4x - c)^2 \]

### Step 3: Expanding the Right Side
Now, we expand the right side:
\[ (x^2 - 4x - c)^2 = x^4 - 8x^3 + (16 + 2c)x^2 - 8cx + c^2 \]
Thus, the equation becomes:
\[ 8x^2 - 32x - 8c = x^4 - 8x^3 + (16 + 2c)x^2 - 8cx + c^2 \]

### Step 4: Rearranging the Equation
Rearranging gives us:
\[ x^4 - 8x^3 + (16 + 2c - 8)x^2 + (8c - 32)x + (c^2 + 8c) = 0 \]
This simplifies to:
\[ x^4 - 8x^3 + (2c + 8)x^2 + (8c - 32)x + (c^2 + 8c) = 0 \]

### Step 5: Finding Conditions for Distinct Real Roots
For the quartic equation to have exactly two distinct real solutions, the discriminant of the quadratic formed by the roots must be greater than zero. We will analyze the conditions for the quadratic formed by the roots of the original equation.

1. **For the quadratic \( x^2 - 4x - c = 0 \)**:
- The discriminant must be greater than zero:
\[ (-4)^2 - 4(1)(-c) > 0 \implies 16 + 4c > 0 \implies c > -4 \]

2. **For the quadratic \( x^2 - 4x + c + 8 = 0 \)**:
- The discriminant must also be greater than zero:
\[ (-4)^2 - 4(1)(c + 8) > 0 \implies 16 - 4(c + 8) > 0 \implies 16 - 4c - 32 > 0 \implies -4c > 16 \implies c < -4 \]

### Step 6: Combining the Conditions
From the two conditions derived:
1. \( c > -4 \)
2. \( c < -12 \)

We find that \( c \) must lie in the interval:
\[ c \in (-12, -4) \]

### Step 7: Finding \( b - a \)
Here, \( a = -12 \) and \( b = -4 \). Therefore:
\[ b - a = -4 - (-12) = -4 + 12 = 8 \]

### Final Answer
The value of \( b - a \) is \( \boxed{8} \).