To determine the excluded values of [tex]\( x \)[/tex] in the rational expression [tex]\( f(x) = \frac{x^2 + 7x}{x^2 + 14x} \)[/tex], we need to identify the values of [tex]\( x \)[/tex] that make the denominator equal to zero. These values are excluded because division by zero is undefined.
Here’s a detailed, step-by-step solution:
1. Identify the denominator of the rational expression:
The denominator of the given rational expression is [tex]\( x^2 + 14x \)[/tex].
2. Set the denominator equal to zero to find the excluded values:
[tex]\[
x^2 + 14x = 0
\][/tex]
3. Solve the equation [tex]\( x^2 + 14x = 0 \)[/tex]:
Factor out the common factor [tex]\( x \)[/tex]:
[tex]\[
x(x + 14) = 0
\][/tex]
This equation implies two possible solutions when set to zero:
[tex]\[
x = 0 \quad \text{or} \quad x + 14 = 0
\][/tex]
4. Solve for the values of [tex]\( x \)[/tex] in each case:
- For [tex]\( x = 0 \)[/tex]:
[tex]\[
x = 0
\][/tex]
- For [tex]\( x + 14 = 0 \)[/tex]:
[tex]\[
x + 14 = 0 \implies x = -14
\][/tex]
5. List the excluded values:
The excluded values are [tex]\( x = 0 \)[/tex] and [tex]\( x = -14 \)[/tex]. These are the values that make the denominator zero, which makes the rational expression undefined.
In conclusion, the excluded values of [tex]\( x \)[/tex] for the given rational expression are:
[tex]\[
x \neq 0, -14
\][/tex]
