If ` ( x^(2) + 1)/( x) = 4 (1)/( 4)` then what is the value of `( x^(3) + (1)/( x^(3)))` ?

Options

A. `(529)/(16)`
B. `(527)/(64)`
C. `(4913)/(64)`
D. `(4097)/(64)`

Answer

To solve the equation \( \frac{x^2 + 1}{x} = 4 \frac{1}{4} \) and find the value of \( x^3 + \frac{1}{x^3} \), we will follow these steps:

### Step 1: Simplify the given equation
We start with the equation:
\[ \frac{x^2 + 1}{x} = 4 \frac{1}{4} \]
This simplifies to:
\[ \frac{x^2 + 1}{x} = \frac{17}{4} \]
Multiplying both sides by \( x \) gives:
\[ x^2 + 1 = \frac{17}{4} x \]

### Step 2: Rearrange the equation
Rearranging the equation, we get:
\[ x^2 - \frac{17}{4}x + 1 = 0 \]

### Step 3: Use the quadratic formula
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -\frac{17}{4}, c = 1 \):
\[ b^2 - 4ac = \left(-\frac{17}{4}\right)^2 - 4 \cdot 1 \cdot 1 = \frac{289}{16} - \frac{64}{16} = \frac{225}{16} \]
Thus,
\[ x = \frac{\frac{17}{4} \pm \sqrt{\frac{225}{16}}}{2} = \frac{\frac{17}{4} \pm \frac{15}{4}}{2} \]
This gives us two possible values for \( x \):
\[ x = \frac{32/4}{2} = 4 \quad \text{or} \quad x = \frac{2/4}{2} = \frac{1}{4} \]

### Step 4: Find \( x + \frac{1}{x} \)
Now, we calculate \( x + \frac{1}{x} \) for both values:
1. For \( x = 4 \):
\[ x + \frac{1}{x} = 4 + \frac{1}{4} = \frac{16}{4} + \frac{1}{4} = \frac{17}{4} \]
2. For \( x = \frac{1}{4} \):
\[ x + \frac{1}{x} = \frac{1}{4} + 4 = \frac{1}{4} + \frac{16}{4} = \frac{17}{4} \]

### Step 5: Calculate \( x^3 + \frac{1}{x^3} \)
Using the identity:
\[ x^3 + \frac{1}{x^3} = \left(x + \frac{1}{x}\right)^3 - 3\left(x + \frac{1}{x}\right) \]
Substituting \( x + \frac{1}{x} = \frac{17}{4} \):
\[ x^3 + \frac{1}{x^3} = \left(\frac{17}{4}\right)^3 - 3\left(\frac{17}{4}\right) \]
Calculating \( \left(\frac{17}{4}\right)^3 \):
\[ \left(\frac{17}{4}\right)^3 = \frac{4913}{64} \]
Calculating \( 3 \cdot \frac{17}{4} \):
\[ 3 \cdot \frac{17}{4} = \frac{51}{4} = \frac{816}{64} \]
Now substituting back:
\[ x^3 + \frac{1}{x^3} = \frac{4913}{64} - \frac{816}{64} = \frac{4913 - 816}{64} = \frac{4097}{64} \]

### Final Answer
Thus, the value of \( x^3 + \frac{1}{x^3} \) is:
\[ \frac{4097}{64} \]