- Factor the denominators: \(x^2-36 = (x+6)(x-6)\) and \(x^2+6x = x(x+6)\).
- Identify \(a=6\).
- Find a common denominator: \(x(x+6)(x-6)\).
- Adjust numerators to get \(b=2\) and \(c=6\).
- Simplify the combined numerator \(2x-(x-6) = x+6\), leading to \(d=2\), \(e=6\), and \(f=6\).
- The identified values are \(a=6, b=2, c=6, d=2, e=6, f=6\).
Explanation
- Factorization and Identifying 'a'
To subtract the rational expressions \(\frac{2}{x^2-36} - \frac{1}{x^2+6x}\), we first factor the denominators.
The first denominator is a difference of squares: \(x^2 - 36 = (x+6)(x-6)\).
The second denominator can be factored by taking out the common factor \(x\): \(x^2 + 6x = x(x+6)\).
Thus, the expression becomes:
\[\frac{2}{(x+6)(x-6)} - \frac{1}{x(x+6)}\]
Comparing this to the template \(\frac{2}{(x+6)(x-6)} - \frac{1}{x(x+a)}\), we identify \(a = 6\).
- Finding Common Denominators and Identifying 'b' and 'c'
To subtract these fractions, we need a common denominator. The least common multiple of \((x+6)(x-6)\) and \(x(x+6)\) is \(x(x+6)(x-6)\).
We adjust the first fraction:
\[\frac{2}{(x+6)(x-6)} = \frac{2 \cdot x}{(x+6)(x-6) \cdot x} = \frac{2x}{x(x+6)(x-6)}\]
Comparing this to \(\frac{bx}{x(x+6)(x-6)}\), we identify \(b = 2\).
Next, we adjust the second fraction:
\[\frac{1}{x(x+6)} = \frac{1 \cdot (x-6)}{x(x+6) \cdot (x-6)} = \frac{x-6}{x(x+6)(x-6)}\]
Comparing this to \(\frac{x-c}{x(x+6)(x-6)}\), we identify \(c = 6\).
- Combining and Simplifying to Identify 'd', 'e', and 'f'
Now we combine the fractions:
\[\frac{2x}{x(x+6)(x-6)} - \frac{x-6}{x(x+6)(x-6)} = \frac{2x - (x-6)}{x(x+6)(x-6)}\]
Expanding the numerator: \(2x - x + 6 = x + 6\).
The template shows \(\frac{dx-x+e}{x(x+6)(x-6)}\).
Matching this to \(\frac{2x - x + 6}{x(x+6)(x-6)}\), we identify \(d = 2\) and \(e = 6\).
Finally, simplifying the expression: \(\frac{x+6}{x(x+6)(x-6)}\).
Comparing this to \(\frac{x+f}{x(x+6)(x-6)}\), we identify \(f = 6\).
(Note: The prompt suggests \(e=12\) and \(f=-6\) in its list, but checking the algebraic steps: \(2x - (x-6) = 2x-x+6 = x+6\). The variable \(e\) corresponds to the constant term in the numerator, which is \(6\). The expression simplifies to \(\frac{x+6}{x(x+6)(x-6)} = \frac{1}{x(x-6)}\).)
Examples
Understanding how to subtract rational expressions is a fundamental skill in algebra, useful for simplifying complex formulas in physics or engineering. For example, when analyzing the total resistance in an electrical circuit involving multiple resistors connected in parallel, you often encounter complex rational expressions that need to be simplified to find a single, manageable value. Mastering this process helps in accurately modeling such systems and predicting their behavior efficiently.