Solve The Equation $x^2 - 4x - 32 = 0$ By Graphing. What Are The Solutions?
The quadratic function is a fundamental concept in algebra, and solving it is a crucial skill in mathematics. Quadratic functions are expressed in the general form ax² + bx + c = 0, where a, b, and c are constants, and x is the variable. The solutions to a quadratic equation, also known as roots or zeros, represent the points where the parabola intersects the x-axis. One effective method for finding these solutions is by graphing the quadratic function. This article delves into the process of solving the quadratic function x² - 4x - 32 = 0 by graphing, providing a step-by-step guide to understanding this approach. By visualizing the function's graph, we can identify the x-intercepts, which directly correspond to the solutions of the equation. This method not only offers a visual representation of the solutions but also enhances our understanding of the behavior of quadratic functions. Understanding how to solve quadratic functions through graphing is essential for various mathematical applications and provides a strong foundation for more advanced algebraic concepts. In this article, we'll break down each step of the graphing process, making it easy to follow and apply to other quadratic equations.
Understanding Quadratic Functions
Before diving into the graphing method, it's essential to understand the basics of quadratic functions. A quadratic function is a polynomial function of degree two, generally represented as f(x) = ax² + bx + c, where a, b, and c are constants, and a is not equal to zero. The graph of a quadratic function is a parabola, a U-shaped curve that opens either upwards or downwards, depending on the sign of the coefficient a. If a is positive, the parabola opens upwards, and if a is negative, it opens downwards. The key features of a parabola include its vertex, axis of symmetry, and x-intercepts (if any). The vertex is the highest or lowest point on the parabola, depending on its orientation. The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The x-intercepts are the points where the parabola intersects the x-axis, and these points represent the real solutions of the quadratic equation ax² + bx + c = 0. When we talk about solving a quadratic equation, we are essentially finding these x-intercepts. Graphing a quadratic function allows us to visually identify these intercepts and, thus, find the solutions to the equation. This method is particularly useful because it provides an intuitive understanding of the roots and the overall behavior of the function. By understanding the relationship between the coefficients a, b, and c and the shape and position of the parabola, we can effectively use graphing to solve quadratic equations. This foundational knowledge is crucial for both solving the given equation x² - 4x - 32 = 0 and for tackling more complex quadratic problems in the future.
Step-by-Step Solution for x² - 4x - 32 = 0 by Graphing
To solve the quadratic equation x² - 4x - 32 = 0 by graphing, we will follow a structured approach. This method involves several steps, each crucial in accurately plotting the parabola and identifying its x-intercepts. These intercepts will then provide the solutions to the equation.
Step 1: Identify the Coefficients
The first step is to identify the coefficients a, b, and c in the quadratic equation. In the equation x² - 4x - 32 = 0, we have a = 1, b = -4, and c = -32. These coefficients play a significant role in determining the shape and position of the parabola. The value of a determines whether the parabola opens upwards (a > 0) or downwards (a < 0). The values of b and c influence the location of the vertex and the y-intercept of the parabola. Correctly identifying these coefficients is essential for the subsequent steps in the graphing process.
Step 2: Find the Vertex
The vertex of the parabola is a critical point as it represents the minimum or maximum value of the quadratic function. The x-coordinate of the vertex can be found using the formula x = -b / 2a. Substituting the values a = 1 and b = -4, we get x = -(-4) / (2 * 1) = 4 / 2 = 2. To find the y-coordinate of the vertex, we substitute x = 2 back into the original equation: y = (2)² - 4(2) - 32 = 4 - 8 - 32 = -36. Therefore, the vertex of the parabola is at the point (2, -36). Knowing the vertex is crucial because it helps us position the parabola on the graph accurately. It also serves as a central reference point around which the rest of the parabola is plotted.
Step 3: Determine the Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two symmetrical halves. The equation of the axis of symmetry is x = h, where h is the x-coordinate of the vertex. In our case, the x-coordinate of the vertex is 2, so the axis of symmetry is the line x = 2. This line is essential for plotting the parabola because it ensures that the graph is symmetrical on both sides of the vertex. When we plot points on one side of the axis of symmetry, we can easily find corresponding points on the other side, simplifying the graphing process.
Step 4: Find the Y-intercept
The y-intercept is the point where the parabola intersects the y-axis. To find the y-intercept, we set x = 0 in the quadratic equation: y = (0)² - 4(0) - 32 = -32. Thus, the y-intercept is at the point (0, -32). The y-intercept provides another key point for plotting the parabola. It helps to determine the vertical position of the parabola and can be particularly useful in sketching the curve when combined with the vertex and other points.
Step 5: Find the X-intercepts (Roots)
The x-intercepts, also known as the roots or solutions of the quadratic equation, are the points where the parabola intersects the x-axis. To find these intercepts, we set y = 0 in the equation x² - 4x - 32 = 0 and solve for x. This can be done by factoring, completing the square, or using the quadratic formula. In this case, we can factor the equation as (x - 8)(x + 4) = 0. Setting each factor to zero gives us x - 8 = 0 or x + 4 = 0. Solving these equations, we find x = 8 and x = -4. Therefore, the x-intercepts are at the points (-4, 0) and (8, 0). These points are the solutions to the quadratic equation and are the primary goal of our graphing method.
Step 6: Plot the Points and Sketch the Graph
Now that we have the vertex (2, -36), the y-intercept (0, -32), and the x-intercepts (-4, 0) and (8, 0), we can plot these points on a coordinate plane. Draw the axis of symmetry as a dashed vertical line at x = 2. Since the parabola is symmetrical about this line, we can plot additional points by reflecting the y-intercept across the axis of symmetry. For example, the point (4, -32) is symmetrical to the y-intercept (0, -32) with respect to the axis of symmetry. With these points plotted, we can sketch the parabola, making sure it passes through all the plotted points smoothly. The parabola should open upwards because the coefficient a = 1 is positive. The graph will visually represent the quadratic function and clearly show the x-intercepts, which are the solutions to the equation.
Step 7: Identify the Solutions
From the graph, we can directly identify the solutions of the quadratic equation as the x-coordinates of the x-intercepts. The parabola intersects the x-axis at x = -4 and x = 8. Therefore, the solutions to the quadratic equation x² - 4x - 32 = 0 are x = -4 and x = 8. This graphical method provides a visual confirmation of the solutions, making it easier to understand the relationship between the quadratic equation and its roots.
Verifying the Solution
To verify the solutions obtained by graphing, we can substitute them back into the original quadratic equation x² - 4x - 32 = 0. This step ensures the accuracy of our solutions and confirms that they satisfy the equation. First, let's substitute x = -4:
(-4)² - 4(-4) - 32 = 16 + 16 - 32 = 32 - 32 = 0
This confirms that x = -4 is indeed a solution. Next, we substitute x = 8:
(8)² - 4(8) - 32 = 64 - 32 - 32 = 64 - 64 = 0
This also confirms that x = 8 is a solution. Since both solutions satisfy the original equation, we can confidently conclude that the solutions x = -4 and x = 8 are correct. Verifying the solutions is an important practice in mathematics, especially when using graphical methods, as it ensures that the visual interpretation is accurate and the algebraic solutions are valid. This step reinforces the understanding of the relationship between the roots of a quadratic equation and its graphical representation.
Conclusion
In conclusion, solving the quadratic equation x² - 4x - 32 = 0 by graphing is a comprehensive method that not only provides the solutions but also enhances our understanding of quadratic functions. By following the step-by-step process, we identified the coefficients, found the vertex and axis of symmetry, determined the y-intercept, and most importantly, located the x-intercepts, which represent the solutions to the equation. The solutions, x = -4 and x = 8, were visually confirmed on the graph and algebraically verified by substituting them back into the original equation. This approach highlights the interconnectedness of algebraic and graphical methods in mathematics. Graphing quadratic functions is not just about finding solutions; it's about visualizing the behavior of these functions and understanding their key characteristics. The ability to solve quadratic equations by graphing is a valuable skill that extends beyond the classroom, finding applications in various fields such as physics, engineering, and economics. Mastering this method lays a solid foundation for more advanced mathematical concepts and problem-solving techniques. This article has provided a detailed guide to solving a specific quadratic equation, but the principles and steps outlined can be applied to any quadratic equation, making it a versatile and powerful tool in mathematical analysis.
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