Solving 2x(x-2) = 200 By Completing The Square

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Solving 2x(x-2) = 200 by Completing the Square

Hey guys! Today, we're diving into a classic algebra problem: solving the quadratic equation 2x(x-2) = 200 by using the completing the square method. This technique is super useful, especially when you want to avoid the quadratic formula or when factoring isn't straightforward. So, let's break it down step by step and make sure we all get it. This method not only helps in solving equations but also provides a deeper understanding of quadratic expressions and their properties.

Understanding the Completing the Square Method

Before we jump into the problem, let's quickly recap what completing the square actually means. Essentially, it's a way to rewrite a quadratic expression in the form of (x + a)^2 + b or (x - a)^2 + b. This form is incredibly handy because it allows us to easily find the solutions (or roots) of the equation. By converting the quadratic equation into this form, we create a perfect square trinomial, which simplifies the process of solving for x. The beauty of this method lies in its ability to transform a seemingly complex equation into a more manageable form. It's like taking a puzzle and rearranging the pieces to see the solution more clearly. Completing the square is not just a mathematical trick; it’s a fundamental technique that enhances our understanding of quadratic equations.

Why Use Completing the Square?

You might be wondering, why bother with completing the square when we have other methods like factoring or the quadratic formula? Well, completing the square offers several advantages. First, it's a reliable method that works for any quadratic equation, even those that are difficult or impossible to factor. Second, it gives us insight into the structure of the quadratic equation, showing us the vertex form of the parabola, which is crucial in graphing and understanding the function's behavior. Lastly, it’s a foundational technique that's used in more advanced math, like calculus. It provides a deeper understanding of mathematical structures and transformations, making it an invaluable tool in your mathematical toolkit. Think of it as learning the fundamentals of a language – once you master it, you can express complex ideas with clarity and precision.

Step-by-Step Solution

Okay, let's get our hands dirty and solve the equation 2x(x-2) = 200. We'll go through each step meticulously to ensure clarity.

Step 1: Simplify the Equation

First, we need to simplify the equation by expanding the left side and moving all terms to one side. This will give us a standard quadratic form, which is essential for applying the completing the square method. Start by distributing the 2x across the parentheses:

2x(x - 2) = 200
2x^2 - 4x = 200

Now, subtract 200 from both sides to set the equation to zero:

2x^2 - 4x - 200 = 0

Step 2: Make the Leading Coefficient 1

To complete the square, the coefficient of the x^2 term must be 1. In our case, it's 2, so we'll divide the entire equation by 2:

(2x^2 - 4x - 200) / 2 = 0 / 2
x^2 - 2x - 100 = 0

This step is crucial because it simplifies the process of finding the term needed to complete the square. It's like setting the stage for the main act, ensuring that all the elements are in place for the transformation.

Step 3: Move the Constant Term to the Right Side

Next, we move the constant term (-100) to the right side of the equation. This isolates the x terms on the left side, preparing us to complete the square:

x^2 - 2x = 100

This separation allows us to focus solely on the x terms and manipulate them to form a perfect square trinomial. It’s like clearing the workspace to focus on the core task at hand.

Step 4: Complete the Square

Here's the heart of the method! To complete the square, we need to add a value to both sides of the equation that will make the left side a perfect square trinomial. This value is calculated by taking half of the coefficient of the x term (which is -2), squaring it, and adding it to both sides. Half of -2 is -1, and (-1)^2 is 1. So, we add 1 to both sides:

x^2 - 2x + 1 = 100 + 1

This step is the essence of the method. By adding the correct value, we create a perfect square trinomial, which can be easily factored. It's like adding the missing piece to a puzzle, completing the picture and revealing the solution.

Step 5: Factor the Left Side

The left side of the equation is now a perfect square trinomial, which can be factored as follows:

(x - 1)^2 = 101

This factorization is the payoff for completing the square. It transforms the equation into a form where we can easily isolate x. It's like unlocking a secret code that reveals the solution.

Step 6: Take the Square Root of Both Sides

To get rid of the square, we take the square root of both sides of the equation. Remember to consider both the positive and negative square roots:

√(x - 1)^2 = ±√101
x - 1 = ±√101

This step is crucial as it reveals the two possible solutions for x. It's like opening a door to two different paths, both leading to a valid answer.

Step 7: Solve for x

Finally, we isolate x by adding 1 to both sides:

x = 1 ± √101

So, the solutions are:

x = 1 + √101
x = 1 - √101

These are the two values of x that satisfy the original equation. It’s like finding the treasure at the end of a long journey, the reward for all the hard work and effort.

Final Answer

Therefore, the solutions to the equation 2x(x-2) = 200 are x = 1 + √101 and x = 1 - √101. We successfully used the completing the square method to find these solutions. It's pretty neat how this method works, right? Each step builds upon the previous one, leading us to the final answer. And there you have it, guys! We've successfully navigated through the process of solving a quadratic equation by completing the square. Remember, practice makes perfect, so try out a few more examples to solidify your understanding. Happy solving!

Key Takeaways

  • Completing the square is a powerful technique for solving quadratic equations.
  • It involves transforming the equation into the form (x + a)^2 + b = 0 or (x - a)^2 + b = 0.
  • The steps include simplifying the equation, making the leading coefficient 1, moving the constant term, completing the square, factoring, taking the square root, and solving for x.
  • This method provides a deeper understanding of quadratic equations and their solutions.
  • Mastering completing the square enhances your problem-solving skills and prepares you for more advanced mathematical concepts.

Practice Problems

To further solidify your understanding, try solving these equations using the completing the square method:

  1. x^2 + 6x + 5 = 0
  2. 2x^2 - 8x + 6 = 0
  3. x^2 + 4x - 3 = 0

Working through these problems will give you valuable practice and help you become more confident in your ability to apply the completing the square method. It’s like going to the gym – the more you exercise your mathematical muscles, the stronger they become!

Conclusion

Completing the square might seem a bit tricky at first, but with practice, it becomes a valuable tool in your mathematical arsenal. It not only helps solve equations but also provides insights into the structure and properties of quadratic functions. Remember to take it one step at a time, and don't hesitate to revisit the steps if you get stuck. Math is like a puzzle, and completing the square is one of the most elegant ways to fit the pieces together. Keep practicing, and you'll be a pro in no time! Solving quadratic equations is a fundamental skill in mathematics, and mastering the completing the square method opens up a world of possibilities in algebra and beyond. So, keep exploring, keep learning, and most importantly, keep enjoying the journey of mathematical discovery!