Final answer:
The student needs to solve for x where f(x) equals g(x) by rearranging the given quadratic equation into standard form and solving for x using the quadratic formula. No solutions are provided due to an inconsistency in the equations mentioned in the question.
Step-by-step explanation:
The student is asking to solve for the values of x for which f(x) = g(x) given the system of equations 3x - 18 = -3x^2 + x + 6 and 19x = -3x^2 + x + 6. The first step is to set the equations equal to each other, which results in the first being a quadratic equation, while the second equation is not necessary for finding x since it is equivalent to the first after simplifying. We will focus on the quadratic equation.
To solve the quadratic equation, we rearrange it into the form
ax^2 + bx + c = 0
by moving all terms to one side:
3x^2 - 4x + 12 = 0
Next, we use the quadratic formula x = (-b ± √(b^2 - 4ac)) / (2a) to find the values of x that satisfy the equation. Substituting the coefficients a = 3, b = -4, and c = 12 into the formula, we calculate the discriminant (b^2 - 4ac), which determines the nature of the roots.