Value of R if
`(a^(2)-19a-25)/(a-7)=a-12+R/(a-7)` is _______.

Options

A. `-109`
B. `-88`
C. `-84`
D. `-64`

Answer

To find the value of \( R \) in the equation

\[ \frac{a^2 - 19a - 25}{a - 7} = a - 12 + \frac{R}{a - 7}, \]

we will follow these steps:

### Step 1: Rearranging the Equation
We start by moving \( a - 12 \) to the left side of the equation:

\[ \frac{a^2 - 19a - 25}{a - 7} - (a - 12) = \frac{R}{a - 7}. \]

### Step 2: Finding a Common Denominator
The left side of the equation can be combined over a common denominator of \( a - 7 \):

\[ \frac{a^2 - 19a - 25 - (a - 12)(a - 7)}{a - 7} = \frac{R}{a - 7}. \]

### Step 3: Expanding the Expression
Now, we need to expand \( (a - 12)(a - 7) \):

\[ (a - 12)(a - 7) = a^2 - 7a - 12a + 84 = a^2 - 19a + 84. \]

### Step 4: Substituting Back into the Equation
Substituting this back into the equation gives:

\[ \frac{a^2 - 19a - 25 - (a^2 - 19a + 84)}{a - 7} = \frac{R}{a - 7}. \]

### Step 5: Simplifying the Numerator
Now simplify the numerator:

\[ a^2 - 19a - 25 - a^2 + 19a - 84 = -25 - 84 = -109. \]

So we have:

\[ \frac{-109}{a - 7} = \frac{R}{a - 7}. \]

### Step 6: Equating the Numerators
Since the denominators are the same, we can equate the numerators:

\[ R = -109. \]

### Final Answer
Thus, the value of \( R \) is

\[ \boxed{-109}. \]

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