To determine the value(s) of [tex]\( x \)[/tex] for which [tex]\( f(x) = g(x) \)[/tex], we need to set the given functions equal to each other and solve for [tex]\( x \)[/tex].
The given functions are:
[tex]\[ f(x) = x^2 + 7x + 33 \][/tex]
[tex]\[ g(x) = -6x - 9 \][/tex]
To find the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = g(x) \)[/tex], set [tex]\( f(x) \)[/tex] equal to [tex]\( g(x) \)[/tex]:
[tex]\[ x^2 + 7x + 33 = -6x - 9 \][/tex]
Next, we move all terms to one side of the equation to set it to zero:
[tex]\[ x^2 + 7x + 33 + 6x + 9 = 0 \][/tex]
Combine like terms:
[tex]\[ x^2 + 13x + 42 = 0 \][/tex]
This is a quadratic equation in the standard form [tex]\( ax^2 + bx + c = 0 \)[/tex]. To solve this quadratic equation, we can factor it. We look for two numbers that multiply to [tex]\( 42 \)[/tex] (the constant term) and add up to [tex]\( 13 \)[/tex] (the coefficient of the [tex]\( x \)[/tex] term).
The two numbers that meet these criteria are [tex]\( 6 \)[/tex] and [tex]\( 7 \)[/tex]:
[tex]\[ (x + 6)(x + 7) = 0 \][/tex]
Setting each factor equal to zero gives us the possible values of [tex]\( x \)[/tex]:
[tex]\[ x + 6 = 0 \quad \text{or} \quad x + 7 = 0 \][/tex]
Solving these equations, we find:
[tex]\[ x = -6 \quad \text{or} \quad x = -7 \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = g(x) \)[/tex] are:
[tex]\[ x = -7 \quad \text{and} \quad x = -6 \][/tex]
