- Combine the fractions using the common denominator \(x^2+2x-8\).
- Simplify the resulting numerator \(2x+8\) to \(2(x+4)\).
- Factor the denominator to \((x+4)(x-2)\).
- Cancel the common factor \((x+4)\) to arrive at \(\boxed{\frac{2}{x-2}}\).
Explanation
- Problem Analysis
The expression given is \(\frac{x^2-x+3}{x^2+2x-8} - \frac{x^2-3x-5}{x^2+2x-8}\). We observe that both terms share the same denominator, \(D = x^2+2x-8\). First, we need to ensure the expression is defined, which requires \(x^2+2x-8
eq 0\). Factoring the quadratic \(x^2+2x-8\), we find it is \((x+4)(x-2)\). Thus, the expression is defined for \(x
eq -4\) and \(x
eq 2\).
- Combining and Simplifying the Numerator
Since the denominators are identical, we can combine the two fractions into a single one by subtracting the numerators:
\[\frac{x^2-x+3 - (x^2-3x-5)}{x^2+2x-8}\]
Next, we distribute the negative sign in the numerator:
\[\frac{x^2-x+3 - x^2 + 3x + 5}{x^2+2x-8}\]
Combining like terms in the numerator:
\[(x^2 - x^2) + (-x + 3x) + (3 + 5) = 0 + 2x + 8 = 2x + 8\]
So the expression simplifies to \(\frac{2x+8}{x^2+2x-8}\).
- Final Simplification
We now factor the numerator and the denominator:
Numerator: \(2x+8 = 2(x+4)\)
Denominator: \(x^2+2x-8 = (x+4)(x-2)\)
Thus, the expression is:
\[\frac{2(x+4)}{(x+4)(x-2)}\]
Assuming \(x
eq -4\), we can cancel the common factor \((x+4)\) from both the numerator and the denominator:
\[\frac{2}{x-2}\]
This matches the first given option.
Examples
This type of simplification is fundamental in algebra, especially when working with rational functions in calculus or physics. For instance, if you are modeling the velocity of an object as the difference between two complex motion equations, simplifying the expression helps in finding critical points or limits (like finding the instantaneous velocity as time approaches a specific value). Being able to reduce complex fractions is an essential skill for solving higher-order equations and analyzing dynamic systems.