To solve the problem, we need to find the value of [tex]\(x\)[/tex] for which [tex]\((f + g)(x) = 0\)[/tex].
First, we need to define the functions:
[tex]\( f(x) = x^2 - 2x \)[/tex]
[tex]\( g(x) = 6x + 4 \)[/tex]
The sum of these functions is:
[tex]\[ (f + g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f + g)(x) = (x^2 - 2x) + (6x + 4) \][/tex]
Next, we combine like terms:
[tex]\[ (f + g)(x) = x^2 - 2x + 6x + 4 \][/tex]
[tex]\[ (f + g)(x) = x^2 + 4x + 4 \][/tex]
Now, we set this expression equal to 0:
[tex]\[ x^2 + 4x + 4 = 0 \][/tex]
This is a quadratic equation that we can solve for [tex]\(x\)[/tex]. We observe that this quadratic is a perfect square:
[tex]\[ x^2 + 4x + 4 = (x + 2)^2 \][/tex]
So, we set the perfect square to zero:
[tex]\[ (x + 2)^2 = 0 \][/tex]
Taking the square root of both sides, we get:
[tex]\[ x + 2 = 0 \][/tex]
Solving for [tex]\(x\)[/tex], we find:
[tex]\[ x = -2 \][/tex]
Thus, the value of [tex]\(x\)[/tex] for which [tex]\((f + g)(x) = 0\)[/tex] is [tex]\(-2\)[/tex].
