Subtract.
$\frac{3}{x^2-10 x+16}-\frac{2}{x-2}$
Simplify your answer as much as possible.
We are given the expression \(\frac{3}{x^2 - 10x + 16} - \frac{2}{x - 2}\). To simplify this, we first need to factor the quadratic expression in the denominator of the first fraction. We are looking for two numbers that multiply to 16 and add to -10. These numbers are -8 and -2. Thus, we have \(x^2 - 10x + 16 = (x-8)(x-2)\). The expression now becomes \(\frac{3}{(x-8)(x-2)} - \frac{2}{x-2}\). Note that for the expression to be defined, \(x eq 8\) and \(x eq 2\).
To subtract the fractions, we need a common denominator. The least common denominator (LCD) is \((x-8)(x-2)\). The first fraction already has this denominator. We multiply the second fraction by \(\frac{x-8}{x-8}\) to get a common denominator:
\[\frac{2}{x-2} = \frac{2(x-8)}{(x-2)(x-8)} = \frac{2x - 16}{(x-8)(x-2)}\]
Now subtract the two fractions:
\[\frac{3}{(x-8)(x-2)} - \frac{2x - 16}{(x-8)(x-2)} = \frac{3 - (2x - 16)}{(x-8)(x-2)}\]
Distribute the negative sign:
\[\frac{3 - 2x + 16}{(x-8)(x-2)} = \frac{19 - 2x}{(x-8)(x-2)}\]
Thus, the simplified expression is:
\[\frac{19 - 2x}{x^2 - 10x + 16}\]
Understanding how to simplify rational expressions is essential in algebra and calculus. For example, if you are analyzing a physical system where a rate of change is modeled by a rational function, simplifying the expression can help you identify key properties like vertical asymptotes (where the denominator is zero) or critical points. This skill is foundational for solving more complex equations, calculating limits, and understanding the behavior of functions in engineering, physics, and economics.