- Factor the denominator \(x^2+4x-5\) as \((x+5)(x-1)\) and the numerator \(x^2+5x\) as \(x(x+5)\).
- Simplify the second term by canceling the common factor \((x+5)\) to obtain \(\frac{x}{x-1}\).
- Find a common denominator \(x(x-1)\) to add the two fractions \(\frac{x+1}{x} + \frac{x}{x-1}\).
- Combine the numerators to reach the final simplified expression: \(\boxed{\frac{2x^2-1}{x(x-1)}}\).
Explanation
- Analyzing and Factoring the Expression
The expression to be simplified is \(\frac{x+1}{x} + \frac{x^2+5x}{x^2+4x-5}\). First, we examine the second fraction, \(\frac{x^2+5x}{x^2+4x-5}\). We begin by factoring both its numerator and denominator. The denominator \(x^2+4x-5\) can be factored as \((x+5)(x-1)\), and the numerator \(x^2+5x\) can be factored as \(x(x+5)\).
- Simplifying the Second Fraction
Substituting the factored forms back into the second fraction, we get:
\[\frac{x(x+5)}{(x+5)(x-1)}\]
Assuming \(x
eq -5\), we can cancel the common factor \((x+5)\) from both the numerator and the denominator, resulting in the simplified form:
\[\frac{x}{x-1}\]
- Adding the Fractions
Now, the original expression can be rewritten as:
\[\frac{x+1}{x} + \frac{x}{x-1}\]
To add these two fractions, we need a common denominator, which is \(x(x-1)\). We adjust the fractions accordingly:
\[\frac{(x+1)(x-1)}{x(x-1)} + \frac{x(x)}{x(x-1)}\]
Expanding the terms in the numerator, we have \((x+1)(x-1) = x^2 - 1\) and \(x(x) = x^2\). Combining them over the common denominator:
\[\frac{(x^2 - 1) + x^2}{x(x-1)} = \frac{2x^2 - 1}{x(x-1)}\]
- Final Result
The simplification leads to the result \(\frac{2x^2 - 1}{x(x-1)}\). Comparing this with the provided options, we see that it matches the first option. The final result of the simplification is \(\boxed{\frac{2x^2-1}{x(x-1)}}\).
Examples
In engineering, especially when dealing with circuits or complex system modeling, we often end up with complicated rational expressions that represent impedance or transfer functions. By simplifying these expressions—just as we did here—we can make the math much easier to handle when solving for variables like resistance or frequency. This practice is essential for verifying circuit stability and optimizing system performance in real-time scenarios.