Find the difference of the following rational expressions.
$\frac{x^2-2}{x^2-5}-\frac{4 x+1}{5-x^2}=\frac{\square}{x^2-5}$
We are given the equation \(\frac{x^2-2}{x^2-5} - \frac{4x+1}{5-x^2} = \frac{\square}{x^2-5}\). Our goal is to identify the numerator that satisfies this equality. We begin by observing the denominators \(x^2-5\) and \(5-x^2\). Note that \(5-x^2\) is equivalent to \(-(x^2-5)\). This observation is crucial because it allows us to express both terms in the subtraction with a common denominator.
Next, we rewrite the second fraction using the common denominator. Since the denominator is \(-(x^2-5)\), we can adjust the sign of the entire fraction:
\[-\frac{4x+1}{5-x^2} = -\frac{4x+1}{-(x^2-5)} = \frac{4x+1}{x^2-5}\]
By substituting this back into our original equation, we can combine the two fractions:
\[\frac{x^2-2}{x^2-5} + \frac{4x+1}{x^2-5} = \frac{(x^2-2) + (4x+1)}{x^2-5}\]
Now, we simplify the numerator by combining the like terms:
\[(x^2 - 2) + (4x + 1) = x^2 + 4x - 2 + 1 = x^2 + 4x - 1\]
Thus, the entire expression becomes \(\frac{x^2 + 4x - 1}{x^2-5}\). Comparing this to our original goal \(\frac{\square}{x^2-5}\), we conclude that the missing numerator is \(x^2 + 4x - 1\).
Rational expressions are used in various fields like physics and engineering, particularly when calculating total resistance in parallel circuits or average speeds. For instance, if you have two parts of a journey with different speed-to-distance relationships, you might need to combine their algebraic representations to find an overall average. Simplifying these expressions as we did here makes complex equations much easier to handle and solve, helping engineers and scientists design more efficient systems.