Questions: Perform the indicated operation. 4/(x^2-6x+5)+1/(x-5)

Solution Steps

To simplify the given expression, we need to find a common denominator. First, factor the quadratic expression in the denominator of the first fraction. Then, rewrite both fractions with the common denominator and combine them.

1. Factor the quadratic expression \(x^2 - 6x + 5\).
2. Rewrite the fractions with a common denominator.
3. Combine the fractions and simplify.

We start with the expression \( x^2 - 6x + 5 \). Factoring this gives us: \[ x^2 - 6x + 5 = (x - 5)(x - 1) \]

The original expression is: \[ \frac{4}{x^2 - 6x + 5} + \frac{1}{x - 5} \] Substituting the factored form, we have: \[ \frac{4}{(x - 5)(x - 1)} + \frac{1}{x - 5} \]

The common denominator for the two fractions is \((x - 5)(x - 1)\). Rewriting the second fraction: \[ \frac{1}{x - 5} = \frac{(x - 1)}{(x - 5)(x - 1)} \] Now, we can combine the fractions: \[ \frac{4}{(x - 5)(x - 1)} + \frac{x - 1}{(x - 5)(x - 1)} = \frac{4 + (x - 1)}{(x - 5)(x - 1)} = \frac{x + 3}{(x - 5)(x - 1)} \]

Final Answer