To determine the excluded values of [tex]\(x\)[/tex] for the rational function [tex]\(\frac{x^2-3x-28}{x^2-2x-35}\)[/tex], we need to find the values of [tex]\(x\)[/tex] that make the denominator equal to zero. A zero denominator would make the function undefined, so these values are excluded from the domain of the function.
The denominator of the rational function is [tex]\(x^2 - 2x - 35\)[/tex]. To find the zeros of the denominator, we need to solve the quadratic equation:
[tex]\[ x^2 - 2x - 35 = 0 \][/tex]
We can solve this quadratic equation by factoring. We look for two numbers that multiply to [tex]\(-35\)[/tex] and add to [tex]\(-2\)[/tex]. These two numbers are [tex]\(-7\)[/tex] and [tex]\(5\)[/tex], because:
[tex]\[ (-7) \times 5 = -35 \][/tex]
[tex]\[ (-7) + 5 = -2 \][/tex]
Using these numbers, we can factor the quadratic equation as follows:
[tex]\[ x^2 - 2x - 35 = (x - 7)(x + 5) = 0 \][/tex]
Setting each factor equal to zero gives us the solutions to the equation:
[tex]\[ x - 7 = 0 \quad \text{or} \quad x + 5 = 0 \][/tex]
[tex]\[ x = 7 \quad \text{or} \quad x = -5 \][/tex]
Therefore, the excluded values of [tex]\(x\)[/tex] are [tex]\(x = -5\)[/tex] and [tex]\(x = 7\)[/tex], as these values make the denominator zero and thus the rational function undefined.
The excluded values are:
[tex]\[ \boxed{x = -5 \text{ and } x = 7} \][/tex]
Given the options provided, the correct answer is:
[tex]\[ x = -5, x = 7 \][/tex]
which corresponds to the option:
[tex]\[ \boxed{x = -5, x = 7} \][/tex]
