To find the value of [tex]\( x \)[/tex] for which [tex]\((f + g)(x) = 0\)[/tex], we first need to understand the given functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
Given:
[tex]\[ f(x) = x^2 - 2x \][/tex]
[tex]\[ g(x) = 6x + 4 \][/tex]
The function [tex]\((f + g)(x)\)[/tex] is simply the sum of [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex]:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
Now, let's substitute [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] into this equation:
[tex]\[ (f + g)(x) = (x^2 - 2x) + (6x + 4) \][/tex]
Combine like terms:
[tex]\[ (f + g)(x) = x^2 - 2x + 6x + 4 \][/tex]
[tex]\[ (f + g)(x) = x^2 + 4x + 4 \][/tex]
We are looking for the value of [tex]\( x \)[/tex] that makes this expression equal to zero:
[tex]\[ x^2 + 4x + 4 = 0 \][/tex]
This is a quadratic equation. To solve it, we can either factor it, complete the square, or use the quadratic formula. Let’s factor it:
The quadratic expression [tex]\( x^2 + 4x + 4 \)[/tex] can be factored as:
[tex]\[ (x + 2)^2 = 0 \][/tex]
To find the roots, we set the factor equal to zero:
[tex]\[ (x + 2)^2 = 0 \][/tex]
[tex]\[ x + 2 = 0 \][/tex]
[tex]\[ x = -2 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] that satisfies [tex]\((f + g)(x) = 0\)[/tex] is:
[tex]\[ x = -2 \][/tex]
Thus, the correct value of [tex]\( x \)[/tex] is [tex]\(-2\)[/tex].
