Questions: Express in simplest form: (x^2-49)/(7x-x^2).

Solution Steps

To simplify the given expression \(\frac{x^{2}-49}{7x-x^{2}}\), we need to factor both the numerator and the denominator. The numerator \(x^2 - 49\) is a difference of squares, and the denominator \(7x - x^2\) can be factored by taking out the common factor.

1. Factor the numerator \(x^2 - 49\) as \((x + 7)(x - 7)\).
2. Factor the denominator \(7x - x^2\) as \(-x(x - 7)\).
3. Simplify the fraction by canceling out the common factor \((x - 7)\).

The numerator \(x^2 - 49\) can be factored as a difference of squares: \[ x^2 - 49 = (x + 7)(x - 7) \]

The denominator \(7x - x^2\) can be rearranged and factored: \[ 7x - x^2 = -x^2 + 7x = -x(x - 7) \]

Now, we can rewrite the original expression: \[ \frac{x^2 - 49}{7x - x^2} = \frac{(x + 7)(x - 7)}{-x(x - 7)} \] We can cancel the common factor \((x - 7)\) from the numerator and denominator (assuming \(x \neq 7\)): \[ = \frac{(x + 7)}{-x} = -\frac{(x + 7)}{x} \]

Final Answer