Finding Roots Of Equations Using Systems Of Equations

In mathematics, finding the roots of an equation is a fundamental problem. Roots, also known as solutions or zeros, are the values of the variable that make the equation true. For polynomial equations, like the one given, $12x^3 - 5x = 2x^2 + x + 6$, finding these roots can be challenging. However, one powerful technique is to rewrite the equation as a system of equations. This method allows us to visualize the problem graphically and leverage the intersection points of the graphs to find the roots. In this detailed explanation, we'll explore how to construct such systems of equations and why they help in root-finding, specifically focusing on the equation $12x^3 - 5x = 2x^2 + x + 6$.

Understanding the Concept of Roots

Before diving into the method of systems of equations, it's crucial to grasp what roots represent. A root of an equation is a value that, when substituted for the variable (in this case, x), satisfies the equation, making both sides equal. Graphically, the roots of an equation f(x) = 0 are the x-coordinates where the graph of y = f(x) intersects the x-axis. For the given cubic equation, we are looking for the x-values that make $12x^3 - 5x$ equal to $2x^2 + x + 6$. These roots can be real numbers, which correspond to actual intersection points on the graph, or complex numbers, which do not have a direct graphical representation on the standard Cartesian plane.

Transforming the Equation

The first step in solving any equation is often to bring all terms to one side, setting the equation equal to zero. This standard form simplifies analysis and makes it easier to apply various solution techniques. For our cubic equation, we rearrange the terms to get:

$12x^3 - 5x - (2x^2 + x + 6) = 0$

Simplifying further:

$12x^3 - 2x^2 - 6x - 6 = 0$

Now, the equation is in the standard form of a cubic polynomial equation. The roots of this equation are the x-values that make the polynomial equal to zero. Finding these roots analytically can be complex, often requiring methods like factoring (if possible), using the Rational Root Theorem, or employing numerical methods. However, the system of equations approach provides a visual and often more intuitive way to understand the roots.

Constructing a System of Equations

The core idea behind using a system of equations is to split the original equation into two separate equations, each representing a different function. The roots of the original equation then correspond to the x-coordinates of the points where the graphs of these two functions intersect. This is because, at the intersection points, the y-values of both functions are equal, which means the original equation is satisfied. There are multiple ways to construct such a system, but the key is to choose functions that are easy to graph and analyze.

For the given equation, $12x^3 - 5x = 2x^2 + x + 6$, a natural way to create a system of equations is to let each side of the equation represent a separate function. This leads to the following system:

$$\begin{cases} y = 12x^3 - 5x \\ y = 2x^2 + x + 6 \end{cases}$$

Here, we have two functions:

1. y = 12x³ - 5x, which is a cubic function.
2. y = 2x² + x + 6, which is a quadratic function.

The roots of the original equation are the x-values where these two graphs intersect. By graphing these functions, we can visually approximate the roots and potentially use numerical methods to find them more precisely.

Why This System Works

The reason this system works lies in the fundamental concept of equality. The original equation, $12x^3 - 5x = 2x^2 + x + 6$, states that for certain values of x, the expressions on both sides are equal. By setting each side equal to y, we are essentially graphing the values of each expression as a function of x. The points of intersection occur where the y-values are the same for both functions, meaning that at those x-values, the expressions $12x^3 - 5x$ and $2x^2 + x + 6$ are equal. These x-values are precisely the roots of the original equation.

Alternative Systems of Equations

While the system we constructed above is the most straightforward, there are other ways to split the equation into a system. For instance, we could rearrange the equation to group different terms and then create the system. However, some systems might not be as intuitive or easy to graph.

Consider the system proposed in option B:

$$\begin{cases} y = 12x^3 - 5x + 6 \\ y = 2x^2 + x \end{cases}$$

This system is derived from a different rearrangement of the original equation. If we subtract $(2x^2 + x)$ from both sides of the original equation, we get:

$12x^3 - 5x - (2x^2 + x) = 6$

Then, rearranging the terms gives:

$12x^3 - 5x = 2x^2 + x + 6$

To match this with the second system, we can rewrite the original equation as:

$12x^3 - 5x - 6 = 2x^2 + x$

If we were to construct a system from this form, we would have:

$$\begin{cases} y = 12x^3 - 5x - 6 \\ y = 2x^2 + x \end{cases}$$

However, this does not match the system in option B. Option B presents the system:

$$\begin{cases} y = 12x^3 - 5x + 6 \\ y = 2x^2 + x \end{cases}$$

This system is incorrect because it does not accurately represent the original equation. Adding 6 to the cubic function changes the equation being solved, and the intersection points of these graphs will not correspond to the roots of the original equation.

Graphical Interpretation

To further illustrate the concept, let's consider the graphs of the functions in the correct system:

1. y = 12x³ - 5x (a cubic function)
2. y = 2x² + x + 6 (a quadratic function)

The graph of y = 12x³ - 5x will have a characteristic S-shape, typical of cubic functions, and will pass through the origin (0, 0). The graph of y = 2x² + x + 6 will be a parabola opening upwards. The points where these two graphs intersect are the solutions to the system of equations. The x-coordinates of these intersection points are the roots of the original cubic equation.

Visualizing these graphs helps in understanding the number and approximate values of the roots. A cubic equation can have up to three real roots, so we would expect to see up to three intersection points. The actual number of real roots depends on the specific coefficients of the polynomial, and some roots may be complex numbers.

Practical Applications

The method of using systems of equations to find roots is not just a theoretical exercise; it has practical applications in various fields, including engineering, physics, and computer science. For example, in engineering design, finding the roots of equations is crucial for determining the stability of systems, optimizing designs, and predicting system behavior. In physics, roots can represent equilibrium points or critical values in a model. In computer graphics and simulations, finding intersection points is a common problem that can be solved using similar techniques.

Moreover, this approach provides a foundation for understanding more advanced numerical methods for root-finding, such as the Newton-Raphson method or bisection method. These methods often use iterative processes to approximate the roots, and the graphical understanding gained from the system of equations approach can help in selecting appropriate initial guesses and interpreting the results.

Conclusion

In conclusion, the correct system of equations to find the roots of the equation $12x^3 - 5x = 2x^2 + x + 6$ is:

$$\begin{cases} y = 12x^3 - 5x \\ y = 2x^2 + x + 6 \end{cases}$$

This system is constructed by setting each side of the original equation equal to y, effectively creating two functions whose intersection points represent the roots. Understanding this method is essential for solving polynomial equations and has broad applications in mathematics and various scientific and engineering disciplines. The key takeaway is that rewriting a single equation as a system allows for a visual and intuitive approach to finding solutions, particularly when dealing with complex equations like cubics. By graphing the functions and identifying their intersection points, we can gain valuable insights into the roots of the original equation and their significance.