If `(x^(2)+6x-16)/(x^(2)-5x+6)=(-6)/(x^(2)-2x-3)`, then which of the following could be a value of x?

Options

A. `-7`
B. `-5`
C. `0`
D. `6`

Answer

To solve the equation

\[ \frac{x^2 + 6x - 16}{x^2 - 5x + 6} = \frac{-6}{x^2 - 2x - 3}, \]

we will follow these steps:

### Step 1: Factor the Quadratic Expressions
First, we need to factor the quadratic expressions in the equation.

1. **Numerator**: \(x^2 + 6x - 16\)
- To factor, we look for two numbers that multiply to \(-16\) and add to \(6\). These numbers are \(8\) and \(-2\).
- Thus, we can write:
\[ x^2 + 6x - 16 = (x + 8)(x - 2). \]

2. **Denominator**: \(x^2 - 5x + 6\)
- We look for two numbers that multiply to \(6\) and add to \(-5\). These numbers are \(-2\) and \(-3\).
- Thus, we can write:
\[ x^2 - 5x + 6 = (x - 2)(x - 3). \]

3. **Right-hand side denominator**: \(x^2 - 2x - 3\)
- We look for two numbers that multiply to \(-3\) and add to \(-2\). These numbers are \(-3\) and \(1\).
- Thus, we can write:
\[ x^2 - 2x - 3 = (x - 3)(x + 1). \]

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

\[ \frac{(x + 8)(x - 2)}{(x - 2)(x - 3)} = \frac{-6}{(x - 3)(x + 1)}. \]

### Step 3: Cancel Common Factors
We can cancel the common factor \((x - 2)\) from both sides (as long as \(x \neq 2\)):

\[ \frac{x + 8}{x - 3} = \frac{-6}{x + 1}. \]

### Step 4: Cross-Multiply
Cross-multiplying gives us:

\[ (x + 8)(x + 1) = -6(x - 3). \]

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

1. Left Side:
\[ x^2 + 9x + 8. \]

2. Right Side:
\[ -6x + 18. \]

So we have:

\[ x^2 + 9x + 8 = -6x + 18. \]

### Step 6: Rearrange the Equation
Bringing all terms to one side gives:

\[ x^2 + 9x + 6x + 8 - 18 = 0, \]
which simplifies to:

\[ x^2 + 15x - 10 = 0. \]

### Step 7: Solve the Quadratic Equation
Now we can use the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}, \]
where \(a = 1\), \(b = 15\), and \(c = -10\).

Calculating the discriminant:

\[ b^2 - 4ac = 15^2 - 4(1)(-10) = 225 + 40 = 265. \]

Now substituting back into the formula:

\[ x = \frac{-15 \pm \sqrt{265}}{2}. \]

### Step 8: Approximate the Roots
Calculating the approximate values:

\[ \sqrt{265} \approx 16.28, \]
thus,

\[ x \approx \frac{-15 + 16.28}{2} \quad \text{or} \quad x \approx \frac{-15 - 16.28}{2}. \]

Calculating these gives:

1. \(x \approx \frac{1.28}{2} \approx 0.64\).
2. \(x \approx \frac{-31.28}{2} \approx -15.64\).

### Final Answer
The possible values of \(x\) are approximately \(0.64\) and \(-15.64\).