Question

lesson 25 review - solving rational equations
3
\\(\frac{x}{x - 2} - \frac{2}{x + 4} = \frac{12}{x^2 + 2x - 8}\\)
1 factor denominators
2 find lcd
3 multiply to create lcd
4 solve using numerator
5 check for extraneous solutions!
multiple - choice question
which denominator can be factored?
\\(x + 4\\)
\\(x - 2\\)
\\(x^2 + 2x - 8\\)
rewatch

Explanation:

Step1: Analyze each denominator

• Denominator \( x + 4 \): It is a linear binomial and cannot be factored further (no two integers multiply to 4 and add to 1 in a way that factors it, as it's already in simplest linear form).
• Denominator \( x - 2 \): Also a linear binomial, cannot be factored further.
• Denominator \( x^2 + 2x - 8 \): This is a quadratic trinomial. We can factor it by finding two numbers that multiply to \(-8\) and add to \(2\). The numbers are \(4\) and \(-2\), so it factors as \((x + 4)(x - 2)\).

Answer:

\( x^2 + 2x - 8 \) (the third option in the multiple - choice, with the label \( x^2 + 2x - 8 \))