Question

select the correct answer.
which quadratic equation is part of the solution process for solving this rational equation using the least common denominator?
\\(\frac{2}{3x - 6} - \frac{1}{3x + 6} = \frac{1}{3}\\)
a. \\(x^2 - x - 2 = 0\\)
b. \\(x^2 - x - 10 = 0\\)
c. \\(x^2 + x - 2 = 0\\)
d. \\(x^2 + x + 10 = 0\\)

Explanation:

Step1: Find the LCD

The denominators are \(3x - 6 = 3(x - 2)\), \(3x + 6 = 3(x + 2)\), and \(3\). The least common denominator (LCD) is \(3(x - 2)(x + 2)=3(x^{2}-4)\).

Step2: Multiply each term by LCD

Multiply each term of the equation \(\frac{2}{3x - 6}-\frac{1}{3x + 6}=\frac{1}{3}\) by \(3(x^{2}-4)\):

• For \(\frac{2}{3(x - 2)}\), multiply by \(3(x^{2}-4)\): \(2(x + 2)\)
• For \(\frac{-1}{3(x + 2)}\), multiply by \(3(x^{2}-4)\): \(-(x - 2)\)
• For \(\frac{1}{3}\), multiply by \(3(x^{2}-4)\): \((x^{2}-4)\)

Step3: Expand and simplify

Expand the left - hand side:
\(2(x + 2)-(x - 2)=x^{2}-4\)
\(2x+4 - x + 2=x^{2}-4\)
\(x + 6=x^{2}-4\)
Rearrange the terms to get a quadratic equation:
\(x^{2}-x - 10 = 0\)

Answer:

B. \(x^{2}-x - 10 = 0\)