Activity 5 Complete Me Solving Quadratic Equations By Completing The Square

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Introduction

In this comprehensive guide, we delve into Activity 5, titled "Complete Me!," which focuses on mastering the technique of completing the square to solve quadratic equations. This method is a cornerstone of algebra, providing a systematic approach to finding the roots of any quadratic equation. We will explore the step-by-step process, illustrating it with detailed solutions to a series of example problems. By understanding the underlying principles and practicing diligently, you can confidently tackle a wide range of quadratic equations and gain a deeper appreciation for their mathematical properties. This article aims to provide a comprehensive and SEO-optimized resource for students and educators alike, ensuring a clear and thorough understanding of completing the square.

Completing the square is a powerful technique used to rewrite a quadratic equation in a form that allows for easy solution. Specifically, we transform the equation into the form (x+a)2=b(x + a)^2 = b, where aa and bb are constants. This form makes it straightforward to find the solutions for xx by taking the square root of both sides. This method is particularly useful when the quadratic equation cannot be easily factored. This article will break down the process into manageable steps, providing a clear understanding of each phase. We will begin with an explanation of the fundamental concepts and then move on to practical examples, ensuring that readers can grasp the method effectively.

Mastering the method of completing the square is not just about finding solutions; it's about gaining a deeper understanding of quadratic equations and their properties. This method connects different aspects of algebra, such as factoring, manipulating expressions, and solving equations. By learning to complete the square, students develop critical problem-solving skills and a stronger foundation in mathematics. Our guide will emphasize not only the mechanics of the method but also the reasoning behind each step, fostering a more intuitive understanding. This holistic approach ensures that readers can apply the method to various problems and contexts with confidence. We will also address common pitfalls and misconceptions, providing clear explanations and strategies to avoid them, thereby ensuring a solid grasp of the technique.

Throughout this guide, we will use a combination of explanations, examples, and practice problems to reinforce your understanding. We encourage you to work through the examples alongside us, paying close attention to each step. Remember, practice is key to mastering any mathematical technique, and completing the square is no exception. By the end of this article, you should feel confident in your ability to solve quadratic equations by completing the square, and you will have a valuable tool for your mathematical toolkit. Our aim is to make this topic accessible and engaging, turning what might seem like a daunting task into a manageable and rewarding learning experience. We believe that with the right guidance and practice, anyone can master completing the square.

Problem 1: Solving $x^2 - 2x = 3$

Let's start with the first quadratic equation: $x^2 - 2x = 3$. Our objective is to solve this equation by completing the square. This involves transforming the left-hand side into a perfect square trinomial. A perfect square trinomial is a trinomial that can be factored into the form (x+a)2(x + a)^2 or (x−a)2(x - a)^2. To achieve this, we need to add a constant to both sides of the equation. The key to finding this constant lies in the coefficient of the xx term.

The coefficient of the xx term in our equation is -2. To find the constant we need to add, we take half of this coefficient and then square it. Half of -2 is -1, and squaring -1 gives us 1. Therefore, we will add 1 to both sides of the equation. This step is crucial in the process of completing the square as it sets up the equation for the next transformation. Adding 1 to both sides ensures that we maintain the equality of the equation while moving closer to our goal of creating a perfect square trinomial. This technique is a fundamental aspect of the method and is consistently applied across different quadratic equations.

Adding 1 to both sides of the equation, we get: $x^2 - 2x + 1 = 3 + 1$. Now, the left-hand side is a perfect square trinomial. It can be factored into (x−1)2(x - 1)^2. On the right-hand side, we have 3+13 + 1, which simplifies to 4. So, our equation now looks like this: $(x - 1)^2 = 4$. This form is much easier to solve because we can simply take the square root of both sides. This transformation is the heart of the completing the square method, turning a complex-looking equation into a straightforward one. The perfect square form allows us to isolate xx and find the solutions relatively easily. Understanding this step is key to mastering the technique.

To solve for xx, we take the square root of both sides of the equation: $\sqrt{(x - 1)^2} = \pm\sqrt{4}$. This gives us two possible equations: x−1=2x - 1 = 2 and x−1=−2x - 1 = -2. Solving the first equation, x−1=2x - 1 = 2, we add 1 to both sides to get x=3x = 3. Solving the second equation, x−1=−2x - 1 = -2, we add 1 to both sides to get x=−1x = -1. Therefore, the solutions to the quadratic equation $x^2 - 2x = 3$ are x=3x = 3 and x=−1x = -1. This final step demonstrates the power of completing the square in providing exact solutions to quadratic equations. By following this method, we can confidently find the roots of any quadratic equation, regardless of its complexity.

Problem 2: Solving $s^2 + 4s - 21 = 0$

Now, let's tackle the second quadratic equation: $s^2 + 4s - 21 = 0$. To solve this by completing the square, we first need to isolate the constant term on the right-hand side of the equation. We do this by adding 21 to both sides, which gives us: $s^2 + 4s = 21$. This step is crucial as it prepares the equation for the subsequent steps in the completing the square process. By moving the constant term, we focus on the quadratic and linear terms, setting the stage for creating a perfect square trinomial. This initial manipulation is a standard practice when applying this method.

Next, we need to find the constant that will complete the square on the left-hand side. The coefficient of the ss term is 4. We take half of this coefficient, which is 2, and then square it, which gives us 4. So, we will add 4 to both sides of the equation. Adding this constant ensures that the left-hand side becomes a perfect square trinomial, a key step in the completing the square method. This process is consistent across different quadratic equations and is based on the algebraic identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2. By adding the appropriate constant, we effectively create the b2b^2 term needed to complete the square.

Adding 4 to both sides, we get: $s^2 + 4s + 4 = 21 + 4$. The left-hand side can now be factored as a perfect square: $(s + 2)^2$. The right-hand side simplifies to 25. Thus, our equation becomes: $(s + 2)^2 = 25$. This transformation is the essence of completing the square, allowing us to rewrite the equation in a more manageable form. The perfect square form makes it easy to solve for ss by taking the square root of both sides. Recognizing and creating this form is a critical skill in mastering this method.

To solve for ss, we take the square root of both sides: $\sqrt{(s + 2)^2} = \pm\sqrt{25}$. This gives us two equations: s+2=5s + 2 = 5 and s+2=−5s + 2 = -5. Solving the first equation, s+2=5s + 2 = 5, we subtract 2 from both sides to get s=3s = 3. Solving the second equation, s+2=−5s + 2 = -5, we subtract 2 from both sides to get s=−7s = -7. Therefore, the solutions to the quadratic equation $s^2 + 4s - 21 = 0$ are s=3s = 3 and s=−7s = -7. This final step highlights the effectiveness of completing the square in finding precise solutions to quadratic equations. By systematically applying the method, we can confidently determine the roots of any quadratic equation, reinforcing the importance of this technique in algebra.

Problem 3: Solving $t^2 + 10t + 9 = 0$

Let's proceed to the third quadratic equation: $t^2 + 10t + 9 = 0$. To solve this equation by completing the square, we first need to move the constant term to the right-hand side. Subtracting 9 from both sides gives us: $t^2 + 10t = -9$. This step is crucial for setting up the equation to complete the square. By isolating the variable terms on one side, we prepare the equation for the transformation into a perfect square trinomial. This initial rearrangement is a fundamental part of the completing the square method.

Now, we need to determine the constant term that will complete the square on the left-hand side. The coefficient of the tt term is 10. We take half of this, which is 5, and square it to get 25. So, we add 25 to both sides of the equation. This addition is the core of completing the square, as it creates the necessary term to form a perfect square trinomial. The principle behind this is based on the algebraic identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2. By adding the appropriate constant, we ensure the left side can be expressed in this squared form.

Adding 25 to both sides of the equation, we have: $t^2 + 10t + 25 = -9 + 25$. The left-hand side is now a perfect square trinomial and can be factored as $(t + 5)^2$. The right-hand side simplifies to 16. So, our equation becomes: $(t + 5)^2 = 16$. This transformation is a key step in completing the square, making the equation easier to solve. By rewriting the equation in this form, we can readily find the values of tt by taking the square root of both sides. This process demonstrates the power of this method in simplifying quadratic equations.

To solve for tt, we take the square root of both sides: $\sqrt{(t + 5)^2} = \pm\sqrt{16}$. This results in two equations: t+5=4t + 5 = 4 and t+5=−4t + 5 = -4. Solving the first equation, t+5=4t + 5 = 4, we subtract 5 from both sides to get t=−1t = -1. Solving the second equation, t+5=−4t + 5 = -4, we subtract 5 from both sides to get t=−9t = -9. Therefore, the solutions to the quadratic equation $t^2 + 10t + 9 = 0$ are t=−1t = -1 and t=−9t = -9. This final step illustrates the effectiveness of completing the square in accurately solving quadratic equations. By following this method, we can systematically determine the roots, reinforcing the importance of this algebraic technique.

Problem 4: Solving $x^2 + 14x + 33 = 0$

Finally, let's address the fourth quadratic equation: $x^2 + 14x + 33 = 0$. As with the previous problems, we will use the method of completing the square to find the solutions. The first step is to isolate the constant term by subtracting 33 from both sides of the equation. This gives us: $x^2 + 14x = -33$. Isolating the constant term is a crucial initial step in the completing the square process. It allows us to focus on the quadratic and linear terms, preparing the equation for the creation of a perfect square trinomial. This rearrangement is a standard technique in this method.

Next, we need to find the value that completes the square on the left-hand side. The coefficient of the xx term is 14. We take half of this coefficient, which is 7, and then square it, which gives us 49. Therefore, we add 49 to both sides of the equation. Adding this constant is the essence of completing the square, as it transforms the left side into a perfect square trinomial. This process is based on the algebraic identity (a+b)2=a2+2ab+b2(a + b)^2 = a^2 + 2ab + b^2. By adding the appropriate constant, we create the b2b^2 term, allowing the expression to be written in the squared form.

By adding 49 to both sides, the equation becomes: $x^2 + 14x + 49 = -33 + 49$. Now, the left-hand side is a perfect square trinomial and can be factored as $(x + 7)^2$. The right-hand side simplifies to 16. So, the equation is now: $(x + 7)^2 = 16$. This transformation is the core of completing the square, simplifying the equation and making it easier to solve. The perfect square form allows us to directly find the values of xx by taking the square root of both sides. This step demonstrates the power of this method in simplifying complex quadratic equations.

To solve for xx, we take the square root of both sides: $\sqrt{(x + 7)^2} = \pm\sqrt{16}$. This gives us two possible equations: x+7=4x + 7 = 4 and x+7=−4x + 7 = -4. Solving the first equation, x+7=4x + 7 = 4, we subtract 7 from both sides to get x=−3x = -3. Solving the second equation, x+7=−4x + 7 = -4, we subtract 7 from both sides to get x=−11x = -11. Therefore, the solutions to the quadratic equation $x^2 + 14x + 33 = 0$ are x=−3x = -3 and x=−11x = -11. This final step showcases the accuracy and effectiveness of completing the square in solving quadratic equations. By systematically applying this method, we can confidently find the roots, reinforcing its importance as a fundamental algebraic technique.

Conclusion

In conclusion, completing the square is a robust and versatile method for solving quadratic equations. Through the step-by-step solutions of the four example problems, we have demonstrated how to transform a quadratic equation into a perfect square form, making it straightforward to find the solutions. This technique is not only a powerful tool for solving equations but also provides a deeper understanding of the structure and properties of quadratic expressions. Mastering this method enhances problem-solving skills and provides a strong foundation for further studies in mathematics.

The process of completing the square involves several key steps: first, isolating the constant term on one side of the equation; second, finding the value needed to complete the square by taking half of the coefficient of the linear term and squaring it; third, adding this value to both sides of the equation; fourth, factoring the perfect square trinomial; and finally, taking the square root of both sides to solve for the variable. Each of these steps is crucial and contributes to the overall effectiveness of the method. By understanding and practicing these steps, one can confidently solve a wide range of quadratic equations.

Throughout this guide, we have emphasized the importance of understanding the underlying principles behind completing the square, rather than merely memorizing the steps. This approach fosters a deeper appreciation for the mathematics involved and enables students to apply the method in various contexts. Additionally, we have highlighted the connections between completing the square and other algebraic concepts, such as factoring and the quadratic formula. These connections help to build a more cohesive understanding of algebra and its applications.

By mastering completing the square, students gain a valuable tool for their mathematical toolkit. This method is not only useful for solving quadratic equations but also serves as a foundation for more advanced topics in algebra and calculus. Furthermore, the problem-solving skills developed through this process are transferable to other areas of mathematics and beyond. Therefore, dedicating time and effort to understanding and practicing completing the square is a worthwhile investment in one's mathematical education. We hope this comprehensive guide has provided you with the knowledge and confidence to tackle quadratic equations using this powerful technique.