To find the values of [tex]\( x \)[/tex] for which the denominator of the function [tex]\( y = \frac{-x+4}{x^2 + 6x + 8} \)[/tex] is equal to zero, we need to determine the roots of the quadratic equation in the denominator.
The denominator of the function is given by:
[tex]\[ x^2 + 6x + 8 \][/tex]
We solve this quadratic equation by finding the values of [tex]\( x \)[/tex] that satisfy:
[tex]\[ x^2 + 6x + 8 = 0 \][/tex]
To solve this, we can either factorize the quadratic or use the quadratic formula. In this case, let's factorize the quadratic expression.
Step-by-step factorization:
1. We look for two numbers that multiply to [tex]\( 8 \)[/tex] (the constant term) and add up to [tex]\( 6 \)[/tex] (the coefficient of the linear term [tex]\( x \)[/tex]).
2. The pairs of factors of [tex]\( 8 \)[/tex] are [tex]\( (1, 8), (2, 4), (-1, -8), (-2, -4) \)[/tex].
3. Among these pairs, the pair [tex]\( (2, 4) \)[/tex] adds up to [tex]\( 6 \)[/tex], while [tex]\( (-2, -4) \)[/tex] adds up to [tex]\( -6 \)[/tex], which is not our pair. The correct pair is [tex]\( 2 \)[/tex] and [tex]\( 4 \)[/tex].
4. Therefore, we write:
[tex]\[ x^2 + 6x + 8 = (x + 2)(x + 4) = 0 \][/tex]
5. The roots of the equation can be found by setting each factor to zero:
[tex]\[ x + 2 = 0 \implies x = -2 \][/tex]
[tex]\[ x + 4 = 0 \implies x = -4 \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which the denominator is equal to zero are:
[tex]\[ x = -2 \][/tex]
[tex]\[ x = -4 \][/tex]
These are the points of discontinuity for the function [tex]\( y = \frac{-x+4}{x^2 + 6x + 8} \)[/tex].
Thus, the correct answer is:
[tex]\[ x = -2, x = -4 \][/tex]
