Solve for x : `(x + 3)/(x - 2) - (1 - x)/(x) = (17)/(4) , x - 0 , 2 `

Answer

To solve the equation

\[ \frac{x + 3}{x - 2} - \frac{1 - x}{x} = \frac{17}{4}, \quad x \neq 0, 2 \]

we will follow these steps:

### Step 1: Find a common denominator
The common denominator for the left-hand side is \(x(x - 2)\). We rewrite the equation:

\[ \frac{(x + 3)x - (1 - x)(x - 2)}{x(x - 2)} = \frac{17}{4} \]

### Step 2: Expand the numerators
Now we expand the numerators:

1. For \((x + 3)x\):
\[ x^2 + 3x \]

2. For \(-(1 - x)(x - 2)\):
\[ -(1 - x)(x - 2) = -(x - 2 - x^2 + 2x) = -(-x^2 + 3x - 2) = x^2 - 3x + 2 \]

Combining these, we have:

\[ x^2 + 3x + x^2 - 3x + 2 = 2x^2 + 2 \]

### Step 3: Set up the equation
Now substituting back into the equation gives:

\[ \frac{2x^2 + 2}{x(x - 2)} = \frac{17}{4} \]

### Step 4: Cross-multiply
Cross-multiplying yields:

\[ 4(2x^2 + 2) = 17x(x - 2) \]

### Step 5: Expand both sides
Expanding both sides:

1. Left side:
\[ 8x^2 + 8 \]

2. Right side:
\[ 17x^2 - 34x \]

So we have:

\[ 8x^2 + 8 = 17x^2 - 34x \]

### Step 6: Rearrange the equation
Rearranging gives:

\[ 0 = 17x^2 - 8x^2 - 34x - 8 \]

This simplifies to:

\[ 9x^2 - 34x - 8 = 0 \]

### Step 7: Factor the quadratic equation
To factor \(9x^2 - 34x - 8\), we look for two numbers that multiply to \(9 \times -8 = -72\) and add to \(-34\). The numbers are \(-36\) and \(2\).

Rewriting the equation:

\[ 9x^2 - 36x + 2x - 8 = 0 \]

Grouping gives:

\[ (9x^2 - 36x) + (2x - 8) = 0 \]

Factoring out common terms:

\[ 9x(x - 4) + 2(x - 4) = 0 \]

Factoring out \((x - 4)\):

\[ (x - 4)(9x + 2) = 0 \]

### Step 8: Solve for \(x\)
Setting each factor to zero gives:

1. \(x - 4 = 0 \Rightarrow x = 4\)
2. \(9x + 2 = 0 \Rightarrow x = -\frac{2}{9}\)

### Final Solution
Thus, the solutions are:

\[ x = 4 \quad \text{and} \quad x = -\frac{2}{9} \]