Questions: Perform the indicated operations, then simplify. 7x^2 / (x^2 - 49) - 7x / (x + 7) + 7 / (x - 7)

Solution Steps

To solve the given expression, we need to perform the indicated operations and simplify the result. We will break down the expression into simpler parts, find a common denominator, and then combine the fractions.

1. Identify the common denominator for the fractions.
2. Rewrite each fraction with the common denominator.
3. Combine the fractions and simplify the resulting expression.

We start with the expression: \[ \frac{7 x^{2}}{x^{2}-49} - \frac{7 x}{x+7} + \frac{7}{x-7} \]

The denominator \(x^{2} - 49\) can be factored as \((x - 7)(x + 7)\). Thus, we rewrite the expression as: \[ \frac{7 x^{2}}{(x - 7)(x + 7)} - \frac{7 x}{x + 7} + \frac{7}{x - 7} \]

The common denominator for the fractions is \((x - 7)(x + 7)\). We rewrite each term: \[ \frac{7 x^{2}}{(x - 7)(x + 7)} - \frac{7 x (x - 7)}{(x + 7)(x - 7)} + \frac{7 (x + 7)}{(x - 7)(x + 7)} \]

Now we combine the fractions: \[ \frac{7 x^{2} - 7 x (x - 7) + 7 (x + 7)}{(x - 7)(x + 7)} \]

Expanding the numerator: \[ 7 x^{2} - 7 x^{2} + 49 x + 7 x + 49 = 56 x + 49 \] Thus, the expression simplifies to: \[ \frac{7(8x + 7)}{(x - 7)(x + 7)} \]

Final Answer