If 'a' is an integer such that `(a+(1)/(a))=(17)/(4)`, then the value of `(a-(1)/(a))` is

Options

A. `4`
B. `(13)/(4)`
C. `(17)/(4)`
D. `(15)/(4)`

Answer

To solve the equation \( a + \frac{1}{a} = \frac{17}{4} \) and find the value of \( a - \frac{1}{a} \), we can follow these steps:

### Step 1: Set up the equation
We start with the equation given in the problem:
\[ a + \frac{1}{a} = \frac{17}{4} \]

### Step 2: Multiply through by \( a \)
To eliminate the fraction, we can multiply both sides of the equation by \( 4a \):
\[ 4a^2 + 4 = 17a \]

### Step 3: Rearrange the equation
Now, rearranging the equation gives us a standard quadratic form:
\[ 4a^2 - 17a + 4 = 0 \]

### Step 4: Solve the quadratic equation
Next, we can use the quadratic formula \( a = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4, b = -17, c = 4 \):
\[ b^2 - 4ac = (-17)^2 - 4 \cdot 4 \cdot 4 = 289 - 64 = 225 \]
\[ \sqrt{225} = 15 \]
Now substituting into the quadratic formula:
\[ a = \frac{17 \pm 15}{8} \]
Calculating the two possible values for \( a \):
1. \( a = \frac{32}{8} = 4 \)
2. \( a = \frac{2}{8} = \frac{1}{4} \)

Since \( a \) must be an integer, we take \( a = 4 \).

### Step 5: Find \( a - \frac{1}{a} \)
Now, we substitute \( a = 4 \) back into the expression we need to evaluate:
\[ a - \frac{1}{a} = 4 - \frac{1}{4} \]
Calculating this gives:
\[ 4 - \frac{1}{4} = \frac{16}{4} - \frac{1}{4} = \frac{15}{4} \]

### Final Answer
Thus, the value of \( a - \frac{1}{a} \) is:
\[ \frac{15}{4} \]