Solve the following equation. `x^4-10x^3+26x^2-10x+1=0`

Answer

To solve the equation \( x^4 - 10x^3 + 26x^2 - 10x + 1 = 0 \), we can follow these steps:

### Step 1: Divide the equation by \( x^2 \)
We start by dividing the entire equation by \( x^2 \):
\[ \frac{x^4}{x^2} - \frac{10x^3}{x^2} + \frac{26x^2}{x^2} - \frac{10x}{x^2} + \frac{1}{x^2} = 0 \]
This simplifies to:
\[ x^2 - 10x + 26 - \frac{10}{x} + \frac{1}{x^2} = 0 \]

### Step 2: Rearrange the equation
We can rearrange the equation to group terms:
\[ x^2 + \frac{1}{x^2} - 10\left(x + \frac{1}{x}\right) + 26 = 0 \]

### Step 3: Substitute \( y = x + \frac{1}{x} \)
Let \( y = x + \frac{1}{x} \). Then, we know that:
\[ x^2 + \frac{1}{x^2} = y^2 - 2 \]
Substituting this into our equation gives:
\[ (y^2 - 2) - 10y + 26 = 0 \]
This simplifies to:
\[ y^2 - 10y + 24 = 0 \]

### Step 4: Solve the quadratic equation
Now we solve the quadratic equation \( y^2 - 10y + 24 = 0 \) using the quadratic formula:
\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -10 \), and \( c = 24 \):
\[ y = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 1 \cdot 24}}{2 \cdot 1} \]
\[ y = \frac{10 \pm \sqrt{100 - 96}}{2} \]
\[ y = \frac{10 \pm \sqrt{4}}{2} \]
\[ y = \frac{10 \pm 2}{2} \]
Thus, we have:
\[ y = \frac{12}{2} = 6 \quad \text{or} \quad y = \frac{8}{2} = 4 \]

### Step 5: Solve for \( x \)
Now we have two cases to solve for \( x \):
1. \( x + \frac{1}{x} = 6 \)
2. \( x + \frac{1}{x} = 4 \)

#### Case 1: \( x + \frac{1}{x} = 6 \)
Multiply through by \( x \):
\[ x^2 - 6x + 1 = 0 \]
Using the quadratic formula:
\[ x = \frac{6 \pm \sqrt{(-6)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{6 \pm \sqrt{36 - 4}}{2} \]
\[ x = \frac{6 \pm \sqrt{32}}{2} \]
\[ x = \frac{6 \pm 4\sqrt{2}}{2} \]
\[ x = 3 \pm 2\sqrt{2} \]

#### Case 2: \( x + \frac{1}{x} = 4 \)
Multiply through by \( x \):
\[ x^2 - 4x + 1 = 0 \]
Using the quadratic formula:
\[ x = \frac{4 \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 1}}{2 \cdot 1} \]
\[ x = \frac{4 \pm \sqrt{16 - 4}}{2} \]
\[ x = \frac{4 \pm \sqrt{12}}{2} \]
\[ x = \frac{4 \pm 2\sqrt{3}}{2} \]
\[ x = 2 \pm \sqrt{3} \]

### Final Solution
Thus, the four roots of the equation \( x^4 - 10x^3 + 26x^2 - 10x + 1 = 0 \) are:
\[ 3 + 2\sqrt{2}, \quad 3 - 2\sqrt{2}, \quad 2 + \sqrt{3}, \quad 2 - \sqrt{3} \]