If `x-(1)/(x)=1`, then the value of `(x^(4)-(1)/(x^(2)))/(3x^(2)+5x-3)` is

Options

A. `(1)/(4)`
B. `(1)/(2)`
C. `(3)/(4)`
D. 0

Answer

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

### Step 1: Solve for \( x \)
Starting with the equation:
\[ x - \frac{1}{x} = 1 \]
Multiply both sides by \( x \) to eliminate the fraction:
\[ x^2 - 1 = x \]
Rearranging gives us:
\[ x^2 - x - 1 = 0 \]
Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 1, b = -1, c = -1 \):
\[ x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
\[ x = \frac{1 \pm \sqrt{1 + 4}}{2} \]
\[ x = \frac{1 \pm \sqrt{5}}{2} \]

### Step 2: Calculate \( x^2 \)
Next, we need to find \( x^2 \):
\[ x^2 = \left(\frac{1 + \sqrt{5}}{2}\right)^2 = \frac{1 + 2\sqrt{5} + 5}{4} = \frac{6 + 2\sqrt{5}}{4} = \frac{3 + \sqrt{5}}{2} \]

### Step 3: Calculate \( x^4 \)
Now, we calculate \( x^4 \):
\[ x^4 = (x^2)^2 = \left(\frac{3 + \sqrt{5}}{2}\right)^2 = \frac{(3 + \sqrt{5})^2}{4} = \frac{9 + 6\sqrt{5} + 5}{4} = \frac{14 + 6\sqrt{5}}{4} = \frac{7 + 3\sqrt{5}}{2} \]

### Step 4: Calculate \( \frac{1}{x^2} \)
Now we find \( \frac{1}{x^2} \):
\[ \frac{1}{x^2} = \frac{2}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{2(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})} = \frac{2(3 - \sqrt{5})}{9 - 5} = \frac{2(3 - \sqrt{5})}{4} = \frac{3 - \sqrt{5}}{2} \]

### Step 5: Calculate \( x^4 - \frac{1}{x^2} \)
Now we can compute \( x^4 - \frac{1}{x^2} \):
\[ x^4 - \frac{1}{x^2} = \frac{7 + 3\sqrt{5}}{2} - \frac{3 - \sqrt{5}}{2} = \frac{(7 + 3\sqrt{5}) - (3 - \sqrt{5})}{2} = \frac{4 + 4\sqrt{5}}{2} = 2 + 2\sqrt{5} \]

### Step 6: Calculate \( 3x^2 + 5x - 3 \)
Next, we calculate \( 3x^2 + 5x - 3 \):
\[ 3x^2 = 3 \cdot \frac{3 + \sqrt{5}}{2} = \frac{9 + 3\sqrt{5}}{2} \]
\[ 5x = 5 \cdot \frac{1 + \sqrt{5}}{2} = \frac{5 + 5\sqrt{5}}{2} \]
Combining these:
\[ 3x^2 + 5x - 3 = \frac{9 + 3\sqrt{5}}{2} + \frac{5 + 5\sqrt{5}}{2} - 3 = \frac{(9 + 5 - 6) + (3\sqrt{5} + 5\sqrt{5})}{2} = \frac{8 + 8\sqrt{5}}{2} = 4 + 4\sqrt{5} \]

### Step 7: Final Calculation
Now, we can find the value of the original expression:
\[ \frac{x^4 - \frac{1}{x^2}}{3x^2 + 5x - 3} = \frac{2 + 2\sqrt{5}}{4 + 4\sqrt{5}} = \frac{2(1 + \sqrt{5})}{4(1 + \sqrt{5})} = \frac{2}{4} = \frac{1}{2} \]

Thus, the final answer is:
\[ \boxed{\frac{1}{2}} \]