Solve for x : `4(x-(1)/(x))^(2)+8(x+(1)/(x))=29`. `x ne 0`.

Answer

To solve the equation \( 4\left(x - \frac{1}{x}\right)^{2} + 8\left(x + \frac{1}{x}\right) = 29 \), we will follow these steps:

### Step 1: Rewrite the equation
Start by rewriting the equation:
\[ 4\left(x - \frac{1}{x}\right)^{2} + 8\left(x + \frac{1}{x}\right) - 29 = 0 \]

### Step 2: Expand the square
Using the identity \( (a - b)^2 = a^2 - 2ab + b^2 \), we can expand \( \left(x - \frac{1}{x}\right)^{2} \):
\[ \left(x - \frac{1}{x}\right)^{2} = x^{2} - 2 + \frac{1}{x^{2}} \]
Thus, we have:
\[ 4\left(x^{2} - 2 + \frac{1}{x^{2}}\right) + 8\left(x + \frac{1}{x}\right) - 29 = 0 \]

### Step 3: Substitute and simplify
Let \( t = x + \frac{1}{x} \). Then, we can express \( x^{2} + \frac{1}{x^{2}} \) in terms of \( t \):
\[ x^{2} + \frac{1}{x^{2}} = t^{2} - 2 \]
Substituting this into the equation gives:
\[ 4(t^{2} - 2) + 8t - 29 = 0 \]
Simplifying:
\[ 4t^{2} - 8 + 8t - 29 = 0 \]
\[ 4t^{2} + 8t - 37 = 0 \]

### Step 4: Solve the quadratic equation
Now we will solve the quadratic equation \( 4t^{2} + 8t - 37 = 0 \) using the quadratic formula:
\[ t = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]
Here, \( a = 4, b = 8, c = -37 \):
\[ t = \frac{-8 \pm \sqrt{8^{2} - 4 \cdot 4 \cdot (-37)}}{2 \cdot 4} \]
Calculating the discriminant:
\[ b^{2} - 4ac = 64 + 592 = 656 \]
Thus:
\[ t = \frac{-8 \pm \sqrt{656}}{8} \]
\[ t = \frac{-8 \pm 8\sqrt{41}}{8} \]
\[ t = -1 \pm \sqrt{41} \]

### Step 5: Find \( x \)
Now we have two cases for \( t \):
1. \( t = -1 + \sqrt{41} \)
2. \( t = -1 - \sqrt{41} \)

For each case, we will solve for \( x \):
\[ x + \frac{1}{x} = t \implies x^{2} - tx + 1 = 0 \]

#### Case 1: \( t = -1 + \sqrt{41} \)
\[ x^{2} - (-1 + \sqrt{41})x + 1 = 0 \]
Using the quadratic formula:
\[ x = \frac{(-(-1 + \sqrt{41})) \pm \sqrt{(-1 + \sqrt{41})^{2} - 4}}{2} \]

#### Case 2: \( t = -1 - \sqrt{41} \)
\[ x^{2} - (-1 - \sqrt{41})x + 1 = 0 \]
Using the quadratic formula:
\[ x = \frac{(-(-1 - \sqrt{41})) \pm \sqrt{(-1 - \sqrt{41})^{2} - 4}}{2} \]

### Final Step: Calculate the roots
Calculating the roots for both cases will yield the final values of \( x \).