Question

posttest: rational relationships
17
enter the correct answer in the box.
consider this rational equation.
\\(\frac{1}{x} + \frac{1}{x - 2} = \frac{1}{4}\\)
use the least common denominator to simplify the rational equation into a standard form quadratic equation.
replace the values of \\(b\\) and \\(c\\) to create the equation.
\\(x^2 - bx + c = 0\\)

Explanation:

Step1: Find the LCD

The denominators are \(x\), \(x - 2\), and \(4\). The least common denominator (LCD) of \(x\), \(x - 2\), and \(4\) is \(4x(x - 2)\).

Step2: Multiply each term by LCD

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

$$
LATEXBLOCK0
$$

Step3: Simplify both sides

Simplify the left - hand side: \(4x-8 + 4x=8x - 8\)
Simplify the right - hand side: \(x^{2}-2x\)
So we have the equation \(8x-8=x^{2}-2x\)

Step4: Rearrange into standard form

Rearrange the equation \(8x - 8=x^{2}-2x\) to the standard quadratic form \(ax^{2}+bx + c = 0\).
Bring all terms to one side: \(x^{2}-2x-8x + 8 = 0\), which simplifies to \(x^{2}-10x + 8 = 0\)

Answer:

For the quadratic equation \(x^{2}-bx + c = 0\), \(b = 10\) and \(c = 8\), so the equation is \(x^{2}-10x + 8 = 0\)