Zeros Of F(x) = 2x^2 + 8x - 3 By Completing The Square

Dec 1, 2025 by Alex Johnson 55 views

Finding Zeros of Quadratic Functions: Completing the Square Method for f(x) = 2x^2 + 8x - 3

Are you grappling with quadratic functions and the completing the square method? Don't worry; you're not alone! Many students find this technique a bit tricky at first, but with a clear explanation and step-by-step approach, it becomes much easier to understand. In this article, we will walk through the process of finding the zeros of the quadratic function f(x) = 2x^2 + 8x - 3 using the completing the square method. Understanding this method is crucial for solving various mathematical problems, from algebra to calculus, and even in real-world applications where quadratic equations pop up. Let's dive in and break it down together, so you can master this essential skill and confidently tackle any similar problems you encounter. Remember, practice makes perfect, so feel free to follow along with your own calculations as we go through each step. By the end of this guide, you'll have a solid grasp of how to apply the completing the square method to find the zeros of a quadratic function, making your math journey a little bit smoother and a lot more rewarding.

Understanding the Completing the Square Method

The completing the square method is a powerful technique used to rewrite a quadratic equation in a form that makes it easier to solve. This method is particularly useful when the quadratic equation cannot be easily factored. The main idea behind completing the square is to transform the quadratic expression into a perfect square trinomial, which can then be expressed as the square of a binomial. This transformation allows us to isolate the variable and find its values, which are the zeros (or roots) of the quadratic function. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants. To apply the completing the square method effectively, it's crucial to have a solid grasp of algebraic manipulations, including factoring, expanding, and simplifying expressions. This method isn't just a mathematical trick; it’s a fundamental tool that provides insights into the structure of quadratic equations and their solutions. Think of it as a way to rearrange and repackage the equation into a more manageable format. By understanding the principles behind completing the square, you'll be better equipped to handle a variety of quadratic problems and gain a deeper appreciation for the elegance of algebra.

Steps Involved in Completing the Square

The process of completing the square involves several key steps that transform the quadratic equation into a solvable form. First, if the coefficient of x^2 (the a term) is not 1, you need to divide the entire equation by this coefficient. This step ensures that the leading coefficient is 1, which is necessary for the subsequent steps. Second, you isolate the constant term (c) on one side of the equation. This means moving the constant term to the other side, leaving the terms with x on one side. Third, you take half of the coefficient of x (the b term), square it, and add it to both sides of the equation. This is the crucial step that creates the perfect square trinomial. The value you add is (b/2)^2. Fourth, you factor the perfect square trinomial into the square of a binomial. This binomial will be in the form (x + b/2)^2 or (x - b/2)^2, depending on the sign of the b term. Fifth, you take the square root of both sides of the equation. Remember to consider both the positive and negative square roots, as both will yield valid solutions. Finally, you solve for x by isolating it on one side of the equation. This usually involves a simple addition or subtraction. By following these steps meticulously, you can successfully complete the square and find the zeros of any quadratic function, regardless of its complexity. Each step plays a vital role in the transformation process, and understanding why each step is necessary will help you master this powerful method.

Applying the Method to f(x) = 2x^2 + 8x - 3

Now, let's apply the completing the square method to the quadratic function f(x) = 2x^2 + 8x - 3. This will provide a concrete example of how to use the steps we discussed earlier. First, we set the function equal to zero to find its zeros: 2x^2 + 8x - 3 = 0. The first step is to ensure that the coefficient of x^2 is 1. Since the coefficient is 2, we divide the entire equation by 2, resulting in x^2 + 4x - 3/2 = 0. Next, we isolate the constant term by adding 3/2 to both sides of the equation, which gives us x^2 + 4x = 3/2. Now comes the crucial step of completing the square. We take half of the coefficient of x, which is 4, so half of it is 2. We then square this value, which gives us 2^2 = 4. We add this value to both sides of the equation: x^2 + 4x + 4 = 3/2 + 4. On the left side, we now have a perfect square trinomial, and on the right side, we simplify the expression. The left side can be factored as (x + 2)^2, and the right side simplifies to 3/2 + 8/2 = 11/2. So our equation becomes (x + 2)^2 = 11/2. We are now ready to take the square root of both sides and solve for x. This methodical approach ensures that we transform the original equation step-by-step into a form that is easy to solve, highlighting the power and elegance of the completing the square method.

Step-by-Step Solution

Let's break down the solution to finding the zeros of f(x) = 2x^2 + 8x - 3 using the completing the square method step-by-step. We start with the equation 2x^2 + 8x - 3 = 0. 1. Divide by the leading coefficient: Divide the entire equation by 2 to make the coefficient of x^2 equal to 1: x^2 + 4x - 3/2 = 0. 2. Isolate the constant term: Add 3/2 to both sides to isolate the terms with x: x^2 + 4x = 3/2. 3. Complete the square: Take half of the coefficient of x (which is 4), square it (2^2 = 4), and add it to both sides: x^2 + 4x + 4 = 3/2 + 4. 4. Factor the perfect square trinomial: Factor the left side as a perfect square and simplify the right side: (x + 2)^2 = 11/2. 5. Take the square root of both sides: Take the square root of both sides, remembering to consider both positive and negative roots: x + 2 = ±√(11/2). 6. Solve for x: Subtract 2 from both sides to isolate x: x = -2 ±√(11/2). Therefore, the zeros of the function are x = -2 + √(11/2) and x = -2 - √(11/2). By meticulously following these steps, we've successfully found the zeros of the quadratic function using the completing the square method. Each step builds upon the previous one, ultimately leading to the solution. This structured approach not only helps in solving the equation but also reinforces the understanding of the method itself. Mastering these steps will equip you to tackle similar quadratic equations with confidence.

The Zeros of the Quadratic Function

After meticulously applying the completing the square method to the quadratic function f(x) = 2x^2 + 8x - 3, we have arrived at the zeros of the function. These zeros, also known as the roots or x-intercepts, are the values of x for which f(x) = 0. From our step-by-step solution, we found that the zeros are x = -2 + √(11/2) and x = -2 - √(11/2). These values represent the points where the parabola, which is the graphical representation of the quadratic function, intersects the x-axis. Understanding the zeros of a quadratic function is crucial in many mathematical and real-world applications. For instance, in physics, they can represent the points at which a projectile hits the ground. In engineering, they can help determine the stability of a system. Graphically, the zeros provide key information about the parabola's position and shape. The two distinct zeros we found indicate that the parabola intersects the x-axis at two different points. If the discriminant (b^2 - 4ac) had been zero, we would have found only one real zero, meaning the parabola would touch the x-axis at only one point. If the discriminant had been negative, we would have found complex zeros, indicating that the parabola does not intersect the x-axis. By knowing the zeros, we can better understand and analyze the behavior of the quadratic function and its applications in various fields. This knowledge forms a fundamental building block for more advanced mathematical concepts and problem-solving.

Conclusion

In conclusion, we've successfully navigated the process of finding the zeros of the quadratic function f(x) = 2x^2 + 8x - 3 using the completing the square method. This method, while initially seeming complex, becomes straightforward when broken down into clear, manageable steps. We began by understanding the core principles of the completing the square technique, which involves transforming a quadratic equation into a perfect square trinomial. We then applied these principles to our specific function, meticulously working through each step: dividing by the leading coefficient, isolating the constant term, completing the square, factoring the perfect square, taking the square root, and finally, solving for x. The zeros we found, x = -2 + √(11/2) and x = -2 - √(11/2), represent the points where the parabola intersects the x-axis. This process not only provides the solutions but also deepens our understanding of quadratic functions and their properties. The ability to find zeros is crucial in various mathematical and real-world applications, making the completing the square method a valuable tool in any mathematician's arsenal. Remember, practice is key to mastering this technique. By working through similar problems, you'll build confidence and proficiency in applying the completing the square method. For further learning and practice, you might find helpful resources on websites like Khan Academy, which offers detailed explanations and exercises on quadratic equations and completing the square.