Sure! Let's find the values of [tex]\( x \)[/tex] for which the expression [tex]\(\frac{x-4}{x^2 + 11x + 18}\)[/tex] is undefined.
The expression is undefined where the denominator is zero because division by zero is not defined in mathematics. So, we need to solve for [tex]\( x \)[/tex] such that the denominator equals zero:
[tex]\[ x^2 + 11x + 18 = 0 \][/tex]
This is a quadratic equation, and we can solve it by factoring. We need to find two numbers that multiply to 18 (the constant term) and add up to 11 (the coefficient of the linear term). The numbers 2 and 9 meet this criteria because:
[tex]\[ 2 \times 9 = 18 \][/tex]
[tex]\[ 2 + 9 = 11 \][/tex]
So, we can factor the quadratic as follows:
[tex]\[ x^2 + 11x + 18 = (x + 2)(x + 9) \][/tex]
Setting each factor equal to zero gives us the solutions:
[tex]\[ x + 2 = 0 \Rightarrow x = -2 \][/tex]
[tex]\[ x + 9 = 0 \Rightarrow x = -9 \][/tex]
Therefore, the expression [tex]\(\frac{x-4}{x^2 + 11x + 18}\)[/tex] is undefined for [tex]\( x = -2 \)[/tex] and [tex]\( x = -9 \)[/tex].
So, the excluded values for [tex]\( x \)[/tex] are:
[tex]\[ x = -2, -9. \][/tex]
