Solve each of the following quadratic equations:
`(2)/(x^(2))-(5)/(x)+2=0`

Explanation

To solve the quadratic equation \( \frac{2}{x^2} - \frac{5}{x} + 2 = 0 \), we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we will multiply the entire equation by \( x^2 \) (the common denominator):

\[ x^2 \left( \frac{2}{x^2} - \frac{5}{x} + 2 \right) = 0 \]

This simplifies to:

\[ 2 - 5x + 2x^2 = 0 \]

### Step 2: Rearrange the equation
Now, we will rearrange the equation in standard quadratic form \( ax^2 + bx + c = 0 \):

\[ 2x^2 - 5x + 2 = 0 \]

### Step 3: Factor the quadratic equation
Next, we will factor the quadratic equation \( 2x^2 - 5x + 2 \). We need to find two numbers that multiply to \( 2 \times 2 = 4 \) (the product of \( a \) and \( c \)) and add to \( -5 \) (the coefficient of \( b \)).

The numbers that satisfy this condition are \( -4 \) and \( -1 \). Thus, we can rewrite the equation as:

\[ 2x^2 - 4x - x + 2 = 0 \]

### Step 4: Group the terms
Now, we will group the terms:

\[ (2x^2 - 4x) + (-x + 2) = 0 \]

### Step 5: Factor by grouping
Now, we can factor out common terms from each group:

\[ 2x(x - 2) - 1(x - 2) = 0 \]

Now, factor out the common binomial factor \( (x - 2) \):

\[ (2x - 1)(x - 2) = 0 \]

### Step 6: Set each factor to zero
Now, we will set each factor equal to zero to find the roots:

1. \( 2x - 1 = 0 \)
- Solving for \( x \):
\[ 2x = 1 \implies x = \frac{1}{2} \]

2. \( x - 2 = 0 \)
- Solving for \( x \):
\[ x = 2 \]

### Final Solution
The roots of the equation are:

\[ x = 2 \quad \text{and} \quad x = \frac{1}{2} \]

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