Subtract.
$\frac{x^2-x+3}{x^2+2 x-8}-\frac{x^2-3 x-5}{x^2+2 x-8}$
The problem asks us to simplify the expression \(\frac{x^2 - x + 3}{x^2 + 2x - 8} - \frac{x^2 - 3x - 5}{x^2 + 2x - 8}\). We observe that both fractions share the same denominator, \(x^2 + 2x - 8\). Therefore, we can combine them into a single fraction by subtracting their numerators: \(\frac{(x^2 - x + 3) - (x^2 - 3x - 5)}{x^2 + 2x - 8}\). Note that the expression is defined only when the denominator is non-zero, i.e., \(x^2 + 2x - 8 eq 0\).
We distribute the negative sign across the second numerator to get the full expression for the numerator: \((x^2 - x + 3) - (x^2 - 3x - 5) = x^2 - x + 3 - x^2 + 3x + 5\). Now, we group and combine like terms: \((x^2 - x^2) + (-x + 3x) + (3 + 5) = 0 + 2x + 8 = 2x + 8\). Thus, the combined fraction is \(\frac{2x + 8}{x^2 + 2x - 8}\).
Next, we factor both the numerator and the denominator. The numerator is \(2x + 8 = 2(x + 4)\). The denominator is a quadratic expression \(x^2 + 2x - 8\), which can be factored as \((x + 4)(x - 2)\). Substituting these back into our expression, we have: \(\frac{2(x + 4)}{(x + 4)(x - 2)}\). Since the factor \((x + 4)\) is present in both the numerator and the denominator, we can cancel it out, provided \(x eq -4\). After cancellation, we are left with \(\frac{2}{x - 2}\), where \(x eq -4\) and \(x eq 2\).
Rational expressions are used in various fields like physics and engineering to model relationships between variables, such as calculating the time taken to travel a distance at varying speeds or analyzing electrical circuit components. Simplifying these expressions allows us to find more efficient ways to compute values and understand the behavior of systems. For example, if you are modeling the efficiency of a system represented by the difference between two complex processes, simplifying that difference can reveal how the system's output changes relative to a single variable \(x\). In our case, the expression \(\frac{x^2 - x + 3}{x^2 + 2x - 8} - \frac{x^2 - 3x - 5}{x^2 + 2x - 8}\) simplifies to \(\frac{2}{x - 2}\), making it much easier to analyze how the result behaves as \(x\) changes.