If `(x^(2) - 3x +2)/( x ^(2) - 5x + 4) = (x ^(2) - 6x + 8)/( x ^(2) - 9x + 14),` then the value of x is

Options

A. ` 2 (1)/(2) `
B. `(1)/(2)`
C. `2`
D. `-2`

Answer

To solve the equation

\[ \frac{x^2 - 3x + 2}{x^2 - 5x + 4} = \frac{x^2 - 6x + 8}{x^2 - 9x + 14}, \]

we will follow these steps:

### Step 1: Factor the Quadratic Expressions
First, we will factor each quadratic expression in the equation.

1. **Numerator of the left side:**
\[ x^2 - 3x + 2 = (x - 1)(x - 2) \]

2. **Denominator of the left side:**
\[ x^2 - 5x + 4 = (x - 1)(x - 4) \]

3. **Numerator of the right side:**
\[ x^2 - 6x + 8 = (x - 2)(x - 4) \]

4. **Denominator of the right side:**
\[ x^2 - 9x + 14 = (x - 7)(x - 2) \]

### Step 2: Rewrite the Equation
Now, substituting the factored forms back into the equation, we get:

\[ \frac{(x - 1)(x - 2)}{(x - 1)(x - 4)} = \frac{(x - 2)(x - 4)}{(x - 7)(x - 2)} \]

### Step 3: Cancel Common Factors
We can cancel the common factors from both sides of the equation, keeping in mind that \(x \neq 1\) and \(x \neq 2\) to avoid division by zero:

\[ \frac{x - 2}{x - 4} = \frac{x - 4}{x - 7} \]

### Step 4: Cross-Multiply
Next, we cross-multiply to eliminate the fractions:

\[ (x - 2)(x - 7) = (x - 4)(x - 4) \]

### Step 5: Expand Both Sides
Now, we will expand both sides of the equation:

1. **Left Side:**
\[ x^2 - 7x - 2x + 14 = x^2 - 9x + 14 \]

2. **Right Side:**
\[ (x - 4)^2 = x^2 - 8x + 16 \]

### Step 6: Set the Equation
Setting both sides equal gives us:

\[ x^2 - 9x + 14 = x^2 - 8x + 16 \]

### Step 7: Simplify the Equation
Subtract \(x^2\) from both sides:

\[ -9x + 14 = -8x + 16 \]

Now, rearranging gives:

\[ -9x + 8x = 16 - 14 \]

This simplifies to:

\[ -x = 2 \]

### Step 8: Solve for x
Thus, we find:

\[ x = -2 \]

### Conclusion
The value of \(x\) is \(-2\).

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