If `(x^2 -x- 6)/(x^2 -4x +3)=4/3` , find x .

Options

A. `-10`
B. `-2`
C. 2
D. 10

Answer

To solve the equation

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

we will follow these steps:

### Step 1: Cross-Multiply
Cross-multiplying gives us:

\[ 3(x^2 - x - 6) = 4(x^2 - 4x + 3). \]

### Step 2: Distribute
Now, we will distribute both sides:

\[ 3x^2 - 3x - 18 = 4x^2 - 16x + 12. \]

### Step 3: Rearrange the Equation
Next, we will move all terms to one side of the equation:

\[ 0 = 4x^2 - 3x^2 - 16x + 3x + 12 + 18. \]

This simplifies to:

\[ 0 = x^2 - 13x + 30. \]

### Step 4: Factor the Quadratic
Now we need to factor the quadratic equation \(x^2 - 13x + 30\). We look for two numbers that multiply to \(30\) and add to \(-13\). These numbers are \(-3\) and \(-10\). Thus, we can write:

\[ (x - 3)(x - 10) = 0. \]

### Step 5: Solve for x
Setting each factor to zero gives us the solutions:

\[ x - 3 = 0 \quad \Rightarrow \quad x = 3, \]
\[ x - 10 = 0 \quad \Rightarrow \quad x = 10. \]

### Step 6: Check the Options
The options given are:
- a) -10
- b) -2
- c) 2
- d) 10

Since \(x = 3\) is not an option, we choose \(x = 10\).

### Final Answer
Thus, the solution to the equation is:

\[ \boxed{10}. \]

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