To determine the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 + 6x + 5 \)[/tex], we need to find the values of [tex]\( x \)[/tex] where the function [tex]\( f(x) \)[/tex] is equal to zero. In other words, we need to solve the equation:
[tex]\[ x^2 + 6x + 5 = 0 \][/tex]
This is a quadratic equation, which we can factor. The general form of a quadratic equation is [tex]\( ax^2 + bx + c \)[/tex]. In this case, [tex]\( a = 1 \)[/tex], [tex]\( b = 6 \)[/tex], and [tex]\( c = 5 \)[/tex].
To factor the quadratic expression, we look for two numbers that multiply to [tex]\( c \)[/tex] (which is 5) and add up to [tex]\( b \)[/tex] (which is 6). These numbers are 5 and 1.
Thus, we can factor the quadratic equation as:
[tex]\[ x^2 + 6x + 5 = (x + 5)(x + 1) \][/tex]
Next, we set each factor equal to zero to find the [tex]\( x \)[/tex]-intercepts:
1. [tex]\( x + 5 = 0 \)[/tex]
Solving this equation, we get:
[tex]\[ x = -5 \][/tex]
2. [tex]\( x + 1 = 0 \)[/tex]
Solving this equation, we get:
[tex]\[ x = -1 \][/tex]
Therefore, the [tex]\( x \)[/tex]-intercepts of the function [tex]\( f(x) = x^2 + 6x + 5 \)[/tex] are [tex]\( x = -5 \)[/tex] and [tex]\( x = -1 \)[/tex].
So, Brianna should use the [tex]\( x \)[/tex]-intercepts [tex]\(-5\)[/tex] and [tex]\(-1\)[/tex] to graph [tex]\( f(x) \)[/tex]. The correct answer is:
-5 and -1
