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A system of equations is a set of two or more equations that have common variables. The goal is to find values for these variables that satisfy all the equations at the same time. In this exercise, you're dealing with a quadratic function, which is defined as \(f(x) = ax^2 + bx + c\). Here, \(a\), \(b\), and \(c\) are the variables you need to find.
Given specific values of the function at different points (like \(f(1) = 3\), \(f(-2) = 15\), and \(f(3) = 5\)), you set up a system of three equations. For each point, substitute the \(x\) value into the function to form these equations.
In this example, the equations are:

• \(a + b + c = 3\)
• \(4a - 2b + c = 15\)
• \(9a + 3b + c = 5\)

These equations must all be true, and solving them together will give you the values of \(a\), \(b\), and \(c\). This process is called solving a system of linear equations.

Algebraic manipulation involves rearranging and simplifying equations in order to solve them. This is a crucial skill when working with systems of equations.
In this exercise, you start with three equations, each containing \(a\), \(b\), \(c\). By strategically eliminating one variable at a time, you can reduce this system to simpler forms.
For instance, subtract the first equation from the others to eliminate \(c\):

• From the second equation: \(3a - 3b = 12\), reducing to \(a - b = 4\).
• From the third equation: \(8a + 2b = 2\), simplifying to \(4a + b = 1\).

This results in a simpler system of two equations with two variables. It's easier to work with. By solving this reduced system, you find the values for \(a\) and \(b\).
After \(a\) and \(b\) are found, substitute those values back into one of the original equations to solve for \(c\). This methodically leads to solving the entire system, demonstrating how algebraic manipulation simplifies complex equations step by step.

Function determination is the process of identifying or finding the specific form of a mathematical function that matches given conditions. In this case, for a quadratic function \(f(x) = ax^2 + bx + c\), you use specific values of \(f(x)\) at particular \(x\) values.
Through solving the system of equations, you determine the coefficients \(a\), \(b\), and \(c\). These coefficients define the shape and position of the quadratic function. Here, based on the solution process, you discovered:

• \(a = 1\)
• \(b = -3\)
• \(c = 5\)

With these values, you construct the quadratic function as \(f(x) = x^2 - 3x + 5\).
Function determination is crucial not just for formulating equations, but also for understanding their behavior in real-world scenarios. By learning how to determine the function, you can predict outcomes and understand the relationships between variables in various applications.