Suppose `f(x) = (x - a) (x - b) - (1)/(2) (b - a)`, where a `b in R` . If the minimum value of f(x) is -8.75 , then |b - a + 1| is equal to _______

Answer

To solve the problem, we start with the function given:

\[ f(x) = (x - a)(x - b) - \frac{1}{2}(b - a) \]

### Step 1: Rewrite the function in standard form
We can expand the function \( f(x) \):

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

### Step 2: Identify the vertex
The function \( f(x) \) is a quadratic function, and its minimum value occurs at the vertex. The x-coordinate of the vertex can be found using the formula:

\[ x = -\frac{B}{2A} = \frac{a + b}{2} \]

### Step 3: Calculate the minimum value
Substituting \( x = \frac{a + b}{2} \) into \( f(x) \):

\[ f\left(\frac{a + b}{2}\right) = \left(\frac{a + b}{2} - a\right)\left(\frac{a + b}{2} - b\right) - \frac{1}{2}(b - a) \]

Calculating the first part:

\[ \frac{a + b}{2} - a = \frac{b - a}{2} \]
\[ \frac{a + b}{2} - b = \frac{a - b}{2} \]

Thus,

\[ f\left(\frac{a + b}{2}\right) = \left(\frac{b - a}{2}\right)\left(\frac{a - b}{2}\right) - \frac{1}{2}(b - a) \]
\[ = -\frac{(b - a)^2}{4} - \frac{1}{2}(b - a) \]

### Step 4: Set the minimum value equal to -8.75
Given that the minimum value of \( f(x) \) is -8.75, we set up the equation:

\[ -\frac{(b - a)^2}{4} - \frac{1}{2}(b - a) = -8.75 \]

### Step 5: Multiply through by -4 to eliminate the fraction
Multiplying the entire equation by -4 gives:

\[ (b - a)^2 + 2(b - a) = 35 \]

### Step 6: Let \( t = b - a \)
Let \( t = b - a \), then we have:

\[ t^2 + 2t - 35 = 0 \]

### Step 7: Solve the quadratic equation
Using the quadratic formula \( t = \frac{-B \pm \sqrt{B^2 - 4AC}}{2A} \):

\[ t = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-35)}}{2 \cdot 1} \]
\[ = \frac{-2 \pm \sqrt{4 + 140}}{2} \]
\[ = \frac{-2 \pm \sqrt{144}}{2} \]
\[ = \frac{-2 \pm 12}{2} \]

Calculating the two possible values for \( t \):

1. \( t = \frac{10}{2} = 5 \)
2. \( t = \frac{-14}{2} = -7 \)

### Step 8: Calculate \( |b - a + 1| \)
Now we need to find \( |b - a + 1| \):

1. If \( b - a = 5 \):
\[ |5 + 1| = |6| = 6 \]

2. If \( b - a = -7 \):
\[ |-7 + 1| = |-6| = 6 \]

Thus, in both cases, we find:

\[ |b - a + 1| = 6 \]

### Final Answer
The value of \( |b - a + 1| \) is \( \boxed{6} \).