Express in simplest form: x2−49/7x−x2

Question

.(1 point)
Responses

−(x+7)x
Start Fraction negative left parenthesis x plus 7 right parenthesis over x End Fraction

−7x
Start Fraction negative 7 over x End Fraction

x+7x
Start Fraction x plus 7 over x End Fraction

−7

Answer 1

To simplify the expression \(\frac{x^2 - 49}{7x - x^2}\), we start by factoring the numerator and the denominator where possible.

1. Factor the numerator: \(x^2 - 49\) is a difference of squares and can be factored as: \[x^2 - 49 = (x - 7)(x + 7)\]

2. Factor the denominator: We can rewrite \(7x - x^2\) as: \[7x - x^2 = - (x^2 - 7x) = - (x(x - 7))\] Thus, \(7x - x^2 = -x(x - 7)\).

Now, substituting these factored forms back into the expression, we have: \[ \frac{(x - 7)(x + 7)}{-x(x - 7)} \]

3. Cancel common factors: The factor \((x - 7)\) appears in both the numerator and the denominator, so we can cancel it out (assuming \(x \neq 7\)): \[ \frac{(x + 7)}{-x} = -\frac{x + 7}{x} \]

Putting it all together, the expression simplifies to: \[ -\frac{x + 7}{x} \]

The simplest form of the original expression is: \[ -\frac{x + 7}{x} \]

So, the correct response is: \(-\frac{(x + 7)}{x}\)