Subtract.
$\frac{3}{x^2-10 x+16}-\frac{2}{x-2}$
Simplify your answer as much as possible.
To simplify the expression \(\frac{3}{x^2 - 10x + 16} - \frac{2}{x - 2}\), we first analyze the structure of the denominators. The first denominator is a quadratic trinomial that can be factored. By looking for two numbers that multiply to 16 and add to -10, we identify -8 and -2. Thus, \(x^2 - 10x + 16 = (x - 8)(x - 2)\). Note that for the expression to be defined, the denominators must not be zero, so \(x eq 8\) and \(x eq 2\).
We now have the expression \(\frac{3}{(x - 8)(x - 2)} - \frac{2}{x - 2}\). To subtract these two rational expressions, we need a common denominator, which is \((x - 8)(x - 2)\). We rewrite the second fraction by multiplying its numerator and denominator by \((x - 8)\): \[\frac{2}{x - 2} = \frac{2(x - 8)}{(x - 2)(x - 8)} = \frac{2x - 16}{(x - 8)(x - 2)}\]
Now we can subtract the two fractions by combining their numerators over the common denominator: \[\frac{3}{(x - 8)(x - 2)} - \frac{2x - 16}{(x - 8)(x - 2)} = \frac{3 - (2x - 16)}{(x - 8)(x - 2)}\] Distribute the negative sign in the numerator: \(3 - 2x + 16 = 19 - 2x\). Thus, the simplified expression is: \[\frac{19 - 2x}{(x - 8)(x - 2)}\] or equivalently \(\frac{19 - 2x}{x^2 - 10x + 16}\).
In real-world engineering or financial modeling, you often need to combine different rates or costs that are represented as rational expressions. For instance, if you are calculating the net efficiency of a system involving two components, where efficiency is modelled by a ratio, simplifying these expressions allows for much easier analysis and computation of the final behavior of the model. By simplifying rational expressions like these, engineers can quickly identify critical values or trends without needing to compute complex fractions for every possible input.