Solving Quadratic Equations Graphically: How to Find the Roots of 2x^2 - 13x - 7 0

Some individuals may argue that graphing a quadratic equation is inefficient for obtaining exact solutions. However, the graphical method is a valuable tool that not only provides approximate answers but also aids in understanding the behavior of the equation. In this article, we will demonstrate the graphical method for solving the equation 2x2 - 13x - 7 0.

Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of the second degree, written in the form ax2 bx c 0. It is characterized by the presence of an x2 term, which distinguishes it from linear equations. Solving quadratic equations can be done using the quadratic formula, which always yields exact solutions, but it can also be done graphically, offering a visual understanding of the solutions.

The Given Equation

Let's revisit the equation 2x2 - 13x - 7 0. The coefficients are a 2, b -13, and c -7. Using the quadratic formula x (-b ± √(b2 - 4ac)) / 2a, we can find the exact solutions. However, for the purposes of this article, we will use the graphical method.

Graphing the Quadratic Equation

The first step is to graph the equation y 2x2 - 13x - 7. This involves plotting the quadratic function to identify the points where the graph intersects the x-axis. These points of intersection are the solutions to the equation, as they represent the x-values for which y 0.

Given that the equation is in the form of a parabola – y ax2 bx c where a 2, the parabola opens upwards (since a > 0). We can use graphing software or a graphing calculator to plot the function and visualize the graph.

Identifying the Solutions

The solutions to the equation 2x2 - 13x - 7 0 are the x-values where the graph of y 2x2 - 13x - 7 intersects the x-axis. By examining the graph, you can determine the approximate x-values of these points.

For a more precise approach, let's proceed with the exact solutions using the quadratic formula:

x (-b ± √(b2 - 4ac)) / 2a

x (-(-13) ± √((-13)2 - 4 * 2 * (-7))) / (2 * 2)

x (13 ± √(169 56)) / 4

x (13 ± √225) / 4

x (13 ± 15) / 4

x1 (13 15) / 4 28 / 4 7

x2 (13 - 15) / 4 -2 / 4 -1/2

Therefore, the exact solutions to the equation 2x2 - 13x - 7 0 are x 7 and x -1/2.

Graphical Method vs. Quadratic Formula

The graphical method provides a visual approach to solving quadratic equations, making it easier to understand the relationship between the equation and its solutions. While the quadratic formula always yields exact solutions, the graphical method offers a valuable supplementary tool. It not only provides an approximate solution but also enhances the understanding of the equation's behavior.

Conclusion: The graphical method for solving quadratic equations, such as 2x2 - 13x - 7 0, is a powerful and intuitive approach. It complements the algebraic methods and provides a deeper insight into the nature of quadratic functions.