Solving $42x = -5x^2 + 27$ With Graphing Calculator

Introduction

In mathematics, solving equations is a fundamental skill. Among the various types of equations, quadratic equations hold a significant place due to their wide range of applications in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of the second degree, generally expressed in the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. To solve these equations, we aim to find the values of x that satisfy the equation, which are also known as the roots or solutions of the equation. There are several methods to solve quadratic equations, including factoring, completing the square, using the quadratic formula, and graphical methods. In this article, we will focus on using graphing calculator tables and graphs to solve the quadratic equation 42x = -5x² + 27. This method provides a visual representation of the equation and its solutions, making it an effective tool for understanding the behavior of quadratic functions.

Understanding Quadratic Equations

Before diving into the solution, let's understand the nature of quadratic equations. As mentioned earlier, a quadratic equation is expressed in the form ax² + bx + c = 0. The graph of a quadratic equation is a parabola, which is a U-shaped curve. The solutions of the equation correspond to the points where the parabola intersects the x-axis, also known as the x-intercepts or roots of the equation. A quadratic equation can have two distinct real roots, one real root (when the parabola touches the x-axis at one point), or no real roots (when the parabola does not intersect the x-axis). The number and nature of the roots depend on the discriminant (Δ), which is given by the formula Δ = b² - 4ac. If Δ > 0, the equation has two distinct real roots; if Δ = 0, the equation has one real root; and if Δ < 0, the equation has no real roots. Understanding these concepts is crucial for interpreting the solutions obtained from graphing calculators.

Setting Up the Equation for Graphing

The given equation is 42x = -5x² + 27. To solve this equation using a graphing calculator, we first need to rewrite it in the standard quadratic form ax² + bx + c = 0. To do this, we rearrange the terms to bring all terms to one side of the equation:

5x² + 42x - 27 = 0

Now, the equation is in the standard form, where a = 5, b = 42, and c = -27. This form allows us to easily input the equation into a graphing calculator and analyze its graph and table of values. When using a graphing calculator, it is essential to set an appropriate viewing window to observe the complete behavior of the parabola. The window should be chosen such that it includes the vertex of the parabola and the x-intercepts, if any. The vertex is the point where the parabola changes direction, and its coordinates can be found using the formulas x = -b/(2a) and y = f(-b/(2a)), where f(x) represents the quadratic function. In this case, the x-coordinate of the vertex is x = -42/(2*5) = -4.2. We will keep this in mind when setting the viewing window on the calculator.

Using a Graphing Calculator Table

One way to solve the equation using a graphing calculator is by utilizing the table function. After rewriting the equation in the standard form, we enter the function y = 5x² + 42x - 27 into the calculator. Then, we set up the table to display values of x and the corresponding values of y. The goal is to find the values of x for which y = 0, as these values represent the solutions of the equation. When setting up the table, it's helpful to start with a range of x values that include the estimated solutions. From our earlier calculation, we know the vertex is at x = -4.2, so we can start the table around this value. We can also use trial and error to adjust the table settings until we find the x-values where the y-values are close to zero.

By scrolling through the table, we look for the x values where the y value is either exactly 0 or very close to 0. If we don't find exact zeros, we look for intervals where the y value changes sign (from positive to negative or vice versa). This indicates that a root lies within that interval. For more accurate solutions, we can adjust the table settings to smaller increments for x. In this case, the table function will reveal that the solutions are approximately x = 0.6 and x = -9. These are the values of x that make the equation 5x² + 42x - 27 equal to zero.

Using a Graphing Calculator Graph

Another method to solve the equation is by graphing the function y = 5x² + 42x - 27 on the graphing calculator. The solutions of the equation are the x-intercepts of the graph, i.e., the points where the parabola intersects the x-axis. After entering the equation into the calculator, we graph it and adjust the viewing window to clearly see the parabola and its intercepts. As mentioned earlier, knowing the vertex location helps in setting an appropriate window. We can use the calculator's zoom and window settings to optimize the view.

Once the graph is displayed, we use the calculator's features to find the x-intercepts. Most graphing calculators have a "zero" or "root" function that allows us to find these points. This function usually requires us to set a left bound, a right bound, and a guess for the root. By selecting a region around each x-intercept, the calculator can accurately determine the x-coordinate where the graph crosses the x-axis. Applying this method to the graph of 5x² + 42x - 27, we find that the parabola intersects the x-axis at approximately x = 0.6 and x = -9. These values match the solutions we found using the table method, reinforcing the accuracy of our results. The graphical method provides a visual confirmation of the solutions, making it easier to understand the relationship between the equation and its roots.

Solutions to the Equation

Using both the graphing calculator table and graph methods, we have found the solutions to the equation 42x = -5x² + 27. The solutions are the values of x that satisfy the equation, which we determined to be approximately x = 0.6 and x = -9. These solutions can be verified by substituting them back into the original equation to check if they make the equation true. Let's verify these solutions:

For x = 0.6:

42(0.6) = -5(0.6)² + 27

25.2 = -5(0.36) + 27

25. 2 = -1.8 + 27

26. 2 = 25.2

For x = -9:

42(-9) = -5(-9)² + 27

-378 = -5(81) + 27

-378 = -405 + 27

-378 = -378

Both solutions, x = 0.6 and x = -9, satisfy the original equation, confirming their correctness. Therefore, the solutions to the equation 42x = -5x² + 27 are x = 0.6 and x = -9.

Conclusion

Solving quadratic equations is a crucial skill in mathematics, and graphing calculators provide powerful tools for finding solutions. By using the table and graph functions of a graphing calculator, we can effectively solve equations like 42x = -5x² + 27. The table method allows us to identify solutions by looking for y values close to zero, while the graphical method provides a visual representation of the solutions as x-intercepts. Both methods complement each other, offering a comprehensive approach to solving quadratic equations. In this case, we found the solutions to be x = 0.6 and x = -9. These techniques are valuable not only for solving equations but also for understanding the behavior of quadratic functions and their applications in real-world scenarios. Mastering these methods enhances our ability to tackle more complex mathematical problems and apply them effectively in various fields.