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College Algebra serves as a foundation for a variety of fields including science, technology, engineering, and mathematics. In tackling problems like the one provided, college algebra encompasses operations with algebraic expressions, solving equations, and understanding functions.

The exercise given requires students to formulate and solve a system of nonlinear equations, a common task in college algebra. Formulating equations from verbal descriptions is a critical skill. In this problem, translating 'the sum of two numbers is 20' and 'their product is 96' into the equations \(x + y = 20\) and \(xy = 96\) utilizes the concepts of variables and algebraic expressions.

Once the system of equations is set up, various algebraic methods can be used to find solutions. Solving such systems is a common objective in college algebra courses that equips students with problem-solving skills for more complex scenarios across different domains.

Quadratic equations are a central topic in algebra and are recognizable by their standard form: \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\).

The exercise presents a situation where the quadratic form emerges from a real-world context. After the substitution method is applied, we obtain a quadratic equation from the equation \(x(20 - x) = 96\), which simplifies to \(x^2 - 20x + 96 = 0\).

To solve it, one might factor the equation, complete the square, or apply the quadratic formula. The solution uses the quadratic formula, a powerful tool that guarantees a solution to any quadratic equation, given by \(x = {[-b ± sqrt{b^2 - 4ac}]} / 2a\). Through this formula, students are introduced to concepts like discriminants and are also exposed to the idea of two possible solutions, leading to a deeper understanding of function graphs and how they intersect the x-axis at these points.

The substitution method is one of several techniques for solving systems of equations that involve replacing variables with equivalent expressions.

It is particularly useful when one equation in the system is solved for one variable in terms of the others. In the given problem, the equation \(x + y = 20\) can be manipulated to express \(y\) as \(y = 20 - x\), isolating one variable.

This expression for \(y\) is then substituted into the second equation, which ultimately leads to a quadratic equation. Substitution is often the preferred method when dealing with nonlinear systems like the one in our exercise, as it reduces the system to a single-variable equation that we can then solve using appropriate techniques. Learning this method enhances problem-solving skills by teaching students to simplify complex problems, a valuable skill across various fields of study.