Simplify the expression: Start Fraction x plus 6 over x squared plus 3 x End Fraction minus Start Fraction 5 over x squared plus x minus 6 End Fraction

Question

(1 point)
Responses

Start Fraction x minus 4 over x squared minus 2 x End Fraction
Image with alt text: Start Fraction x minus 4 over x squared minus 2 x End Fraction

Start Fraction x minus 1 over x plus 6 End Fraction
Image with alt text: Start Fraction x minus 1 over x plus 6 End Fraction

Start Fraction x minus 1 over x cubed plus 3 x squared minus 2 x End Fraction
Image with alt text: Start Fraction x minus 1 over x cubed plus 3 x squared minus 2 x End Fraction

Start Fraction x squared plus 9 x minus 12 over x cubed plus 3 x squared minus 2 x End Fraction

Answer 1

To simplify the expression

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

we will first factor the denominators.

1. Factor the denominators:
   • \(x^2 + 3x = x(x + 3)\)
   • \(x^2 + x - 6\) factors to \((x - 2)(x + 3)\).

Now we rewrite the expression with the factored denominators:

\[ \frac{x + 6}{x(x + 3)} - \frac{5}{(x - 2)(x + 3)} \]

2. Find a common denominator: The common denominator of \(x(x + 3)\) and \((x - 2)(x + 3)\) is \(x(x - 2)(x + 3)\).

3. Rewrite each fraction with the common denominator: \[ \frac{(x + 6)(x - 2)}{x(x - 2)(x + 3)} - \frac{5x}{x(x - 2)(x + 3)} \]

4. Combine the fractions: \[ \frac{(x + 6)(x - 2) - 5x}{x(x - 2)(x + 3)} \]

5. Expand the numerator: \[ (x + 6)(x - 2) = x^2 - 2x + 6x - 12 = x^2 + 4x - 12 \] So now the expression is: \[ \frac{x^2 + 4x - 12 - 5x}{x(x - 2)(x + 3)} = \frac{x^2 - x - 12}{x(x - 2)(x + 3)} \]

6. Factor the numerator \(x^2 - x - 12\): This can be factored to \((x - 4)(x + 3)\), so we have: \[ \frac{(x - 4)(x + 3)}{x(x - 2)(x + 3)} \]

7. Cancel the common factors: The \((x + 3)\) cancels out, giving us: \[ \frac{x - 4}{x(x - 2)} \]

Thus, the final simplified expression is: \[ \frac{x - 4}{x^2 - 2x} \]

So the correct answer among your provided options is:

\(\frac{x - 4}{x^2 - 2x}\)