Solve the equation `6x^4-35x^3+62x^2-35x+6=0` .

Answer

To solve the equation \( 6x^4 - 35x^3 + 62x^2 - 35x + 6 = 0 \), we will follow these steps:

### Step 1: Divide the equation by \( x^2 \)
We start by dividing the entire equation by \( x^2 \) (assuming \( x \neq 0 \)):
\[ 6x^2 - 35x + 62 - \frac{35}{x} + \frac{6}{x^2} = 0 \]
This simplifies to:
\[ 6x^2 - 35x + 62 - 35\frac{1}{x} + 6\frac{1}{x^2} = 0 \]

### Step 2: Rewrite the equation
We can rewrite the equation in terms of \( y = x + \frac{1}{x} \):
\[ 6\left(x^2 + \frac{1}{x^2}\right) - 35\left(x + \frac{1}{x}\right) + 62 = 0 \]
Notice that:
\[ x^2 + \frac{1}{x^2} = \left(x + \frac{1}{x}\right)^2 - 2 = y^2 - 2 \]
Thus, we can substitute:
\[ 6(y^2 - 2) - 35y + 62 = 0 \]
This simplifies to:
\[ 6y^2 - 35y + 50 = 0 \]

### Step 3: Solve the quadratic equation
Now we will use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 6 \), \( b = -35 \), and \( c = 50 \).
\[ y = \frac{35 \pm \sqrt{(-35)^2 - 4 \cdot 6 \cdot 50}}{2 \cdot 6} \]
Calculating the discriminant:
\[ (-35)^2 = 1225 \]
\[ 4 \cdot 6 \cdot 50 = 1200 \]
\[ 1225 - 1200 = 25 \]
Thus, we have:
\[ y = \frac{35 \pm 5}{12} \]
Calculating the two values:
1. \( y_1 = \frac{40}{12} = \frac{10}{3} \)
2. \( y_2 = \frac{30}{12} = \frac{5}{2} \)

### Step 4: Solve for \( x \)
Now we have two equations:
1. \( x + \frac{1}{x} = \frac{10}{3} \)
2. \( x + \frac{1}{x} = \frac{5}{2} \)

#### For \( x + \frac{1}{x} = \frac{10}{3} \):
Multiplying by \( 3x \):
\[ 3x^2 - 10x + 3 = 0 \]
Using the quadratic formula:
\[ x = \frac{10 \pm \sqrt{(-10)^2 - 4 \cdot 3 \cdot 3}}{2 \cdot 3} \]
Calculating the discriminant:
\[ 100 - 36 = 64 \]
Thus:
\[ x = \frac{10 \pm 8}{6} \]
Calculating the roots:
1. \( x_1 = \frac{18}{6} = 3 \)
2. \( x_2 = \frac{2}{6} = \frac{1}{3} \)

#### For \( x + \frac{1}{x} = \frac{5}{2} \):
Multiplying by \( 2x \):
\[ 2x^2 - 5x + 2 = 0 \]
Using the quadratic formula:
\[ x = \frac{5 \pm \sqrt{(-5)^2 - 4 \cdot 2 \cdot 2}}{2 \cdot 2} \]
Calculating the discriminant:
\[ 25 - 16 = 9 \]
Thus:
\[ x = \frac{5 \pm 3}{4} \]
Calculating the roots:
1. \( x_3 = \frac{8}{4} = 2 \)
2. \( x_4 = \frac{2}{4} = \frac{1}{2} \)

### Final Solutions
The solutions to the equation \( 6x^4 - 35x^3 + 62x^2 - 35x + 6 = 0 \) are:
\[ x = 3, \quad x = \frac{1}{3}, \quad x = 2, \quad x = \frac{1}{2} \]

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