To determine the value of [tex]\( x \)[/tex] for which [tex]\( (f+g)(x) = 0 \)[/tex], we should first express [tex]\( (f+g)(x) \)[/tex] by combining 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]
We combine these to find [tex]\( (f+g)(x) \)[/tex]:
[tex]\[ (f+g)(x) = f(x) + g(x) \][/tex]
[tex]\[ (f+g)(x) = (x^2 - 2x) + (6x + 4) \][/tex]
[tex]\[ (f+g)(x) = x^2 - 2x + 6x + 4 \][/tex]
[tex]\[ (f+g)(x) = x^2 + 4x + 4 \][/tex]
Now, we need to solve the equation [tex]\( (f+g)(x) = 0 \)[/tex]:
[tex]\[ x^2 + 4x + 4 = 0 \][/tex]
This is a quadratic equation. To solve it, we can factorize it or use the quadratic formula. In this case, the quadratic can be factorized as:
[tex]\[ x^2 + 4x + 4 = (x + 2)(x + 2) \][/tex]
[tex]\[ x^2 + 4x + 4 = (x + 2)^2 \][/tex]
Setting the equation to zero:
[tex]\[ (x + 2)^2 = 0 \][/tex]
Taking the square root of both sides:
[tex]\[ x + 2 = 0 \][/tex]
Solving for [tex]\( x \)[/tex]:
[tex]\[ x = -2 \][/tex]
Therefore, the value of [tex]\( x \)[/tex] for which [tex]\( (f+g)(x) = 0 \)[/tex] is:
[tex]\[ \boxed{-2} \][/tex]
