Solve each of the following equatins :
`(2)/x^(2)-(5)/(x)+2=0,xne0`

Answer

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

### Step 1: Eliminate the fractions by multiplying through by \( x^2 \)
We start by multiplying every term in the equation by \( x^2 \) to eliminate the denominators:

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

This simplifies to:

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

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

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

### Step 3: Factor the quadratic equation
Now we will factor the quadratic equation. 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 are \( -4 \) and \( -1 \).

We can rewrite the equation as:

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

### Step 4: Group the terms
Next, we group the terms:

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

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

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

### Step 6: Factor out the common binomial
We can factor out \( (x - 2) \):

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

### Step 7: Set each factor to zero
Now we set each factor equal to zero:

1. \( 2x - 1 = 0 \)
2. \( x - 2 = 0 \)

### Step 8: Solve for \( x \)
Solving these equations gives us:

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

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

### Final Answer
Thus, the solutions to the equation are:

\[ x = \frac{1}{2}, \quad x = 2 \]

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