Simplify the following rational expressions.
$\frac{5 x}{x^2+5 x+6}+\frac{x}{x+3}=\frac{\square}{x^2+5 x+6}$
We are tasked with simplifying the sum of two rational expressions: \[\frac{5x}{x^2 + 5x + 6} + \frac{x}{x + 3}\] and expressing the result with the common denominator \(x^2 + 5x + 6\). First, note that the quadratic expression \(x^2 + 5x + 6\) can be factored as \((x+2)(x+3)\). For the expressions to be well-defined, we assume \(x eq -2\) and \(x eq -3\).
To combine the two rational expressions, we need them to share the common denominator \(x^2 + 5x + 6\). The first term \(\frac{5x}{x^2 + 5x + 6}\) already has this denominator. For the second term \(\frac{x}{x + 3}\), we can multiply both the numerator and the denominator by \((x+2)\) to create the common denominator: \[\frac{x}{x+3} = \frac{x(x+2)}{(x+3)(x+2)} = \frac{x^2 + 2x}{x^2 + 5x + 6}\]
Now that both terms share the same denominator, we can add them by summing their numerators: \[\frac{5x}{x^2 + 5x + 6} + \frac{x^2 + 2x}{x^2 + 5x + 6} = \frac{5x + x^2 + 2x}{x^2 + 5x + 6}\] Combining the like terms in the numerator (the \(x\) terms), we get \(5x + 2x = 7x\). Thus, the numerator becomes \(x^2 + 7x\). The complete expression is: \[\frac{x^2 + 7x}{x^2 + 5x + 6}\] Comparing this to our original target form \(\frac{\square}{x^2 + 5x + 6}\), we identify the missing expression in the box.
Understanding how to simplify rational expressions is essential for solving complex engineering problems and physics equations where variables need to be grouped together. For example, when calculating the total resistance in an electrical circuit or the center of mass in a mechanical system, you often encounter multiple fractions that must be combined using a common denominator to find a single, simplified expression for further analysis. Mastering this technique allows for cleaner and more efficient mathematical modeling.