Question

lesson 25 review - solving rational equations
③ \\(\frac{x}{x - 2} - \frac{2}{x + 4} = \frac{12}{x^2 + 2x - 8}\\)
\\(lcd = (x + 4)(x - 2)\\) ① factor denominators
② find lcd
③ multiply to create lcd
④ solve using numerator
⑤ check for extraneous solutions!
\\(\frac{x}{(x - 2)} - \frac{2}{(x + 4)} = \frac{12}{(x + 4)(x - 2)}
\frac{x}{(x - 2)}(x + 4) - \frac{2}{(x + 4)}(x - 2) = \frac{12}{(x + 4)(x - 2)}
\\(x(x + 4) - 2(x - 2) = 12\\)
\\(x^2 + 4x - 2x + 4 = 12\\)
\\(x^2 + 2x - 8 = 0\\)
( x
multiple - choice question
what are the factors that will come out of this guess and check step?
□ \\(x - 8\\) and \\(x + 4\\)
□ \\(x - 2\\) and \\(x + 4\\)
□ \\(x - 4\\) and \\(x - 2\\)
□ \\(x - 4\\) and \\(x + 2\\)
□ \\(x - 6\\) and \\(x + 4\\)

Explanation:

Step1: Factor the quadratic denominator

The denominator \(x^2 + 2x - 8\) can be factored. We need two numbers that multiply to \(-8\) and add to \(2\). Those numbers are \(4\) and \(-2\). So, \(x^2 + 2x - 8=(x + 4)(x - 2)\).

Step2: Analyze the guess and check step (factoring the quadratic)

In the equation, after factoring, we see the denominators are \(x - 2\), \(x + 4\), and \((x + 4)(x - 2)\). The step of factoring \(x^2 + 2x - 8\) would yield the factors \(x - 2\) and \(x + 4\) (since \(x^2+2x - 8=(x + 4)(x - 2)\)).

Answer:

B. \(x - 2\) and \(x + 4\)