To find the value(s) of [tex]\( x \)[/tex] for which [tex]\( f(x) = g(x) \)[/tex], we need to set the two functions equal to each other and solve for [tex]\( x \)[/tex].
We start with the given functions:
[tex]\[ f(x) = x^2 + x - 19 \][/tex]
[tex]\[ g(x) = 2x - 7 \][/tex]
Setting [tex]\( f(x) \)[/tex] equal to [tex]\( g(x) \)[/tex], we have:
[tex]\[ x^2 + x - 19 = 2x - 7 \][/tex]
Next, we need to move all terms to one side of the equation to set it to zero. This allows us to solve the quadratic equation:
[tex]\[ x^2 + x - 19 - (2x - 7) = 0 \][/tex]
[tex]\[ x^2 + x - 19 - 2x + 7 = 0 \][/tex]
[tex]\[ x^2 - x - 12 = 0 \][/tex]
Now, we need to solve the quadratic equation [tex]\( x^2 - x - 12 = 0 \)[/tex]. To do this, we can factor the quadratic expression. We look for two numbers that multiply to [tex]\(-12\)[/tex] and add up to [tex]\(-1\)[/tex].
The factors are [tex]\( -4 \)[/tex] and [tex]\( 3 \)[/tex] because:
[tex]\[ -4 \cdot 3 = -12 \][/tex]
[tex]\[ -4 + 3 = -1 \][/tex]
Thus, we can factor the quadratic equation as:
[tex]\[ (x - 4)(x + 3) = 0 \][/tex]
Now, we solve for [tex]\( x \)[/tex] by setting each factor equal to zero:
[tex]\[ x - 4 = 0 \quad \text{or} \quad x + 3 = 0 \][/tex]
Solving these equations, we get:
[tex]\[ x = 4 \quad \text{or} \quad x = -3 \][/tex]
Therefore, the values of [tex]\( x \)[/tex] for which [tex]\( f(x) = g(x) \)[/tex] are:
[tex]\[ x = -3 \quad \text{and} \quad x = 4 \][/tex]
