To find the roots of the equation [tex]\(\frac{x}{x-2}+\frac{1}{x-6}=\frac{4}{x^2-8x+12}\)[/tex] using a graphing tool, we need to follow these steps:
1. Set the equation equal to 0 by subtracting the right side from the left side:
[tex]\[
\frac{x}{x-2} + \frac{1}{x-6} - \frac{4}{x^2-8x+12} = 0
\][/tex]
2. Replace 0 with [tex]\( y \)[/tex]:
[tex]\[
y = \frac{x}{x-2} + \frac{1}{x-6} - \frac{4}{x^2-8x+12}
\][/tex]
3. Input this equation into the graphing tool.
Next, you need to study the graph produced by this equation to find the points where the graph intersects the x-axis. The x-values of these points are the roots of the equation because they make [tex]\( y = 0 \)[/tex].
In this case, the root is:
[tex]\[
-1
\][/tex]
Therefore, the correct answer is:
- [tex]$-1$[/tex]
This means that [tex]\(-1\)[/tex] is the only root of the equation among the given options. The remaining options [tex]\(-0.5\)[/tex], [tex]\(0\)[/tex], [tex]\(2\)[/tex], and [tex]\(6\)[/tex] are not roots of the equation.
