Finding Roots System Of Equations For 2x^3 + 4x^2 - X + 5 = -3x^2 + 4x + 9

In the realm of algebra, determining the roots of an equation is a fundamental task. Roots, also known as solutions or zeros, are the values of the variable that make the equation true. For polynomial equations, which involve terms with variables raised to integer powers, finding roots can sometimes be challenging. However, a powerful technique involves transforming a single equation into a system of equations, which can then be solved graphically or algebraically. In this article, we will explore how to transform the equation $2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9$ into an equivalent system of equations to find its roots.

Understanding Roots and Equations

Before diving into the transformation, let's clarify the concept of roots and equations. An equation is a mathematical statement that asserts the equality of two expressions. For example, $2x + 3 = 7$ is an equation where the left-hand side ($2x + 3$) is equal to the right-hand side ($7$). The variable in the equation, denoted by $x$ in this case, represents an unknown value. The root of an equation is the value of the variable that satisfies the equation, making the left-hand side equal to the right-hand side. Finding roots is equivalent to solving the equation.

For polynomial equations, the roots correspond to the points where the graph of the polynomial function intersects the x-axis. These points are also known as the x-intercepts of the graph. For instance, consider the quadratic equation $x^2 - 5x + 6 = 0$. The roots of this equation are $x = 2$ and $x = 3$, which means that the graph of the quadratic function $y = x^2 - 5x + 6$ intersects the x-axis at the points $(2, 0)$ and $(3, 0)$.

Transforming the Equation into a System

Now, let's focus on the given equation: $2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9$. Our goal is to transform this single equation into a system of two equations. A system of equations is a set of two or more equations that are considered together. The solution to a system of equations is the set of values for the variables that satisfy all equations in the system simultaneously.

To transform the given equation into a system, we can follow these steps:

1. Isolate the terms on one side: First, we need to rearrange the equation so that all terms are on one side, leaving zero on the other side. To do this, we can add $3x^2$, subtract $4x$, and subtract $9$ from both sides of the equation:

   $2x^3 + 4x^2 - x + 5 + 3x^2 - 4x - 9 = -3x^2 + 4x + 9 + 3x^2 - 4x - 9$

   This simplifies to:

   $2x^3 + 7x^2 - 5x - 4 = 0$

2. Define two functions: Next, we can define two functions, $y_1$ and $y_2$, based on the equation. We can set $y_1$ equal to the left-hand side of the equation and $y_2$ equal to the right-hand side, which is zero in this case:

   $y_1 = 2x^3 + 7x^2 - 5x - 4$ $y_2 = 0$

3. Form the system of equations: Now, we can form the system of equations by equating the two functions to $y$:

   $y = 2x^3 + 7x^2 - 5x - 4$ $y = 0$

This system of equations represents the original equation in a different form. The roots of the original equation are the x-values where the graphs of the two equations in the system intersect. In this case, the graph of $y = 2x^3 + 7x^2 - 5x - 4$ intersects the x-axis (which is the graph of $y = 0$) at the roots of the equation.

Alternative Transformation

Another way to transform the original equation into a system is to split the equation into two parts based on the terms present. This method can be useful when the equation has distinct terms that can be grouped separately.

1. Identify two expressions: In the given equation, $2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9$, we can identify two expressions:

   • Expression 1: $2x^3 + 4x^2 - x + 5$
   • Expression 2: $-3x^2 + 4x + 9$

2. Define two functions: We can define two functions, $y_1$ and $y_2$, based on these expressions:

   $y_1 = 2x^3 + 4x^2 - x + 5$ $y_2 = -3x^2 + 4x + 9$

3. Form the system of equations: Now, we can form the system of equations by equating each function to $y$:

   $y = 2x^3 + 4x^2 - x + 5$ $y = -3x^2 + 4x + 9$

In this system, the roots of the original equation are the x-values where the graphs of the two equations intersect. This approach can be particularly useful when visualizing the solutions graphically, as it allows us to see the intersection points of two distinct curves.

Analyzing the Provided Options

Now, let's analyze the options provided in the question to determine which system of equations is equivalent to the original equation.

The original equation is $2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9$. We have already demonstrated two methods for transforming this equation into a system of equations.

Option 1:

$egin{cases} y = 2x^3 + x^2 + 3x + 5 \ y = 9

\end{cases}$

This option is incorrect. To see why, we can set the two expressions for $y$ equal to each other:

$2x^3 + x^2 + 3x + 5 = 9$

Subtracting 9 from both sides, we get:

$2x^3 + x^2 + 3x - 4 = 0$

This equation is not equivalent to the original equation $2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9$, which simplifies to $2x^3 + 7x^2 - 5x - 4 = 0$.

Option 2:

$egin{cases} y = 2x^3 + x^2 \ y = 3x + 14

\end{cases}$

This option is also incorrect. Setting the two expressions for $y$ equal to each other:

$2x^3 + x^2 = 3x + 14$

Rearranging the terms, we get:

$2x^3 + x^2 - 3x - 14 = 0$

This equation is not equivalent to the original equation $2x^3 + 7x^2 - 5x - 4 = 0$.

Option 3:

$egin{cases} y = 2x^3 + 4x^2 - x + 5 \ y = -3x^2 + 4x + 9

\end{cases}$

This option is the correct one. This system of equations directly corresponds to the alternative transformation method we discussed earlier. We defined $y_1$ as $2x^3 + 4x^2 - x + 5$ and $y_2$ as $-3x^2 + 4x + 9$, and then formed the system by equating each function to $y$. The x-values where the graphs of these two equations intersect are the roots of the original equation.

Conclusion

Transforming a single equation into a system of equations is a valuable technique for finding its roots. By defining two functions based on the equation and forming a system, we can solve the equation graphically or algebraically. In the case of the equation $2x^3 + 4x^2 - x + 5 = -3x^2 + 4x + 9$, the equivalent system of equations is:

$egin{cases} y = 2x^3 + 4x^2 - x + 5 \ y = -3x^2 + 4x + 9

\end{cases}$

This system represents the original equation in a different form, and the solutions to the system correspond to the roots of the original equation. Understanding how to transform equations into systems is a crucial skill in algebra and can simplify the process of finding solutions. Remember, the key is to manipulate the equation in a way that allows you to define two separate functions, which then form the basis of your system of equations. By mastering this technique, you'll be well-equipped to tackle a wide range of algebraic problems and find the roots of complex equations with greater ease.