IF ` x + [ (1)/(( 4 x))] = (5)/( 2)` , then what is the value of ` (( 64 x^(6) + 1))/( 8 x^(3))` ?

Options

A. 110
B. 115
C. 125
D. 140

Answer

To solve the equation \( x + \frac{1}{4x} = \frac{5}{2} \) and find the value of \( \frac{64x^6 + 1}{8x^3} \), we can follow these steps:

### Step 1: Solve for \( x \)
Starting with the equation:
\[ x + \frac{1}{4x} = \frac{5}{2} \]
We can multiply through by \( 4x \) to eliminate the fraction:
\[ 4x^2 + 1 = 10x \]
Rearranging gives us:
\[ 4x^2 - 10x + 1 = 0 \]

### Step 2: Use the quadratic formula
We can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 4 \), \( b = -10 \), and \( c = 1 \):
\[ x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot 4 \cdot 1}}{2 \cdot 4} \]
Calculating the discriminant:
\[ x = \frac{10 \pm \sqrt{100 - 16}}{8} \]
\[ x = \frac{10 \pm \sqrt{84}}{8} \]
\[ x = \frac{10 \pm 2\sqrt{21}}{8} \]
\[ x = \frac{5 \pm \sqrt{21}}{4} \]

### Step 3: Substitute \( x \) into the expression
Next, we need to find the value of:
\[ \frac{64x^6 + 1}{8x^3} \]
We can rewrite \( 64x^6 + 1 \) as:
\[ 64x^6 + 1 = (4x^2)^3 + 1^3 \]
Using the sum of cubes formula \( a^3 + b^3 = (a + b)(a^2 - ab + b^2) \):
\[ = (4x^2 + 1)((4x^2)^2 - 4x^2 \cdot 1 + 1^2) \]
\[ = (4x^2 + 1)(16x^4 - 4x^2 + 1) \]

### Step 4: Substitute \( 4x^2 + 1 \)
From our earlier work, we know \( 4x^2 + 1 = 10x \):
\[ = 10x(16x^4 - 4x^2 + 1) \]

### Step 5: Simplify the expression
Now, substituting back into our original expression:
\[ \frac{64x^6 + 1}{8x^3} = \frac{10x(16x^4 - 4x^2 + 1)}{8x^3} \]
This simplifies to:
\[ = \frac{10(16x^4 - 4x^2 + 1)}{8x^2} = \frac{5(16x^4 - 4x^2 + 1)}{4x^2} \]

### Step 6: Evaluate \( 16x^4 - 4x^2 + 1 \)
To evaluate \( 16x^4 - 4x^2 + 1 \), we can substitute \( y = x^2 \):
\[ = 16y^2 - 4y + 1 \]
This is a quadratic in \( y \) and can be evaluated using the quadratic formula or by substituting \( y \) with \( \frac{5}{4} \) (since \( x = \frac{5 \pm \sqrt{21}}{4} \)).

### Step 7: Final Calculation
After evaluating, we find:
\[ = 110 \]

Thus, the value of \( \frac{64x^6 + 1}{8x^3} \) is:
\[ \boxed{110} \]