In your role as an operations analyst, you are evaluating a production cost model governed by the equation ${}18a^2 - 30 = -33a$. Recall the standard mathematical procedure for solving this quadratic equation by factoring. Arrange the following procedural steps in the correct chronological order.

A logistics coordinator is analyzing a resource allocation model represented by the equation ${}18a^2 - 30 = -33a$. After rearranging the equation into the standard form ${}18a^2 + 33a - 30 = 0$, the coordinator must identify the greatest common factor (GCF) to simplify the expression before factoring. According to the standard procedure for solving this equation, what is the GCF that should be factored out?

In a production planning model, an operations analyst uses the quadratic equation ${}18a^2 - 30 = -33a$ to determine the optimal batch size, $a$. To verify the mathematical accuracy of the model, match each component of the solution process with its corresponding mathematical expression.

An operations manager is solving the quadratic equation ${}18a^2 - 30 = -33a$ to optimize supply chain routes. After rearranging and factoring the equation, the manager arrives at the form ${}3(3a - 2)(2a + 5) = 0$. True or False: To find the solutions for $a$ using the Zero Product Property, the manager must set the constant factor 3 equal to zero.

Standardizing a Quadratic Cost Model

A small business owner uses a profit optimization model where the weekly revenue balance factor, $a$, is determined by the quadratic equation ${}18a^2 - 30 = -33a$. To identify the ideal price adjustment, the owner must solve this equation for $a$. Since the balance factor must be positive for the model to be valid, the owner needs to find the positive solution.

What is the positive solution for $a$? Enter your answer as a simplified fraction in the format numerator/denominator (for example, 2/3): ____.

Standard Operating Procedure for Solving Quadratic Optimization Models