Hey everyone! Let's dive into solving a quadratic equation using the completing the square method. We've got a fun one here: 8x² - 48x = -104. Our mission is to transform this equation into a perfect square trinomial, making it much easier to solve for x. So, grab your thinking caps, and let's get started!
Step 1: Making 'a' Equal to 1
In the given quadratic equation, 8x² - 48x = -104, the coefficient of the x² term (which we often call 'a') is 8. But to complete the square effectively, we need 'a' to be 1. How do we do that, you ask? Simple! We divide both sides of the equation by 8. This ensures that we maintain the equation's balance while achieving our goal. Dividing each term by 8, we get:
(8x² / 8) - (48x / 8) = (-104 / 8)
This simplifies to:
x² - 6x = -13
Now, we have a much cleaner equation where the coefficient of x² is 1. This is a crucial step because it sets the stage for the completing the square process. This step is like preparing the foundation before building a house; it ensures that everything else we do is built on a solid base. We now have our equation in the desired form, and we're ready to move on to the next step. Remember, keeping the equation balanced is key, so whatever operation we perform on one side, we must perform on the other. This principle is fundamental in algebra and ensures that our solutions remain accurate. This transformation not only simplifies the equation but also brings us closer to the ultimate solution by making the subsequent steps more manageable and straightforward. It's like decluttering a room before starting a major cleaning project – it makes the whole process more efficient and less daunting. So, with 'a' now equal to 1, we're well-equipped to tackle the next phase of our mathematical journey.
Step 2: Completing the Square
Now comes the exciting part – completing the square! This technique allows us to rewrite one side of our equation as a perfect square trinomial, which is a trinomial that can be factored into the square of a binomial. Our equation currently looks like this: x² - 6x = -13. To complete the square, we need to add a specific constant to both sides of the equation. This constant is calculated by taking half of the coefficient of our x term (which is -6), squaring it, and then adding the result to both sides.
So, let's calculate this magical number. Half of -6 is -3, and (-3)² is 9. That's our constant! Now, we add 9 to both sides of the equation:
x² - 6x + 9 = -13 + 9
This gives us:
x² - 6x + 9 = -4
Notice that the left side of the equation, x² - 6x + 9, is now a perfect square trinomial. It can be factored into (x - 3)². This is the heart of the completing the square method – transforming a quadratic expression into a squared binomial. The beauty of this step lies in its ability to simplify the equation dramatically. By creating a perfect square trinomial, we've essentially transformed a complex quadratic expression into a more manageable form. It's like turning a tangled mess of wires into a neatly organized bundle. This simplification is crucial for solving the equation because it allows us to isolate x more easily. Now, our equation has a squared term, which means we can use the square root property to find the values of x. This step is not just about adding a number; it's about strategically manipulating the equation to reveal its underlying structure and make it solvable. The constant we added acts as a key, unlocking the equation's potential and paving the way for us to find the solutions. So, with our square beautifully completed, we're ready to move on to the final stages of solving this quadratic puzzle.
Step 3: Factoring and Simplifying
As we discussed in the previous step, the left side of our equation, x² - 6x + 9, is a perfect square trinomial. This means it can be factored into (x - 3)². So, our equation now looks like this:
(x - 3)² = -4
This is a significant simplification! We've transformed the quadratic expression into a squared binomial, making it much easier to solve for x. Now, we can move on to the next step, which involves taking the square root of both sides of the equation. This step allows us to undo the squaring operation and isolate the binomial term. The beauty of this factorization lies in its ability to condense a complex expression into a simple, manageable form. It's like compressing a large file into a smaller, more portable package. This simplification not only makes the equation easier to solve but also provides a clear path towards finding the values of x. By factoring the perfect square trinomial, we've essentially unlocked the equation's structure and revealed the relationship between x and the constants. This step is a testament to the power of algebraic manipulation and the elegance of mathematical transformations. The squared binomial is now our key to unlocking the solutions, and we're well on our way to finding them. This transformation is not just a mathematical trick; it's a powerful technique that allows us to solve a wide range of quadratic equations. So, with our equation neatly factored and simplified, we're ready to take the next step and extract the solutions.
Step 4: Taking the Square Root
Now, let's take the square root of both sides of our equation: (x - 3)² = -4. Remember, when we take the square root, we need to consider both the positive and negative roots. This is crucial because both the positive and negative square roots, when squared, will give us the original number.
So, taking the square root of both sides gives us:
√(x - 3)² = ±√(-4)
This simplifies to:
x - 3 = ±√(-4)
Now, we encounter something interesting: the square root of a negative number. This means our solutions will involve imaginary numbers. The square root of -4 can be written as 2i, where i is the imaginary unit (√-1). So, we have:
x - 3 = ±2i
This step is a critical juncture in our solution process. Taking the square root allows us to isolate the binomial term and move closer to finding the values of x. However, the presence of the square root of a negative number introduces the concept of imaginary solutions, expanding our understanding of the possible solutions to quadratic equations. This is a reminder that mathematics can lead us into unexpected territories, where we encounter numbers that go beyond the realm of real numbers. The imaginary unit i is a powerful tool that allows us to work with these solutions and express them in a meaningful way. This step is not just about applying a mathematical operation; it's about embracing the full spectrum of numbers and acknowledging the existence of solutions that may not be immediately apparent. So, with the square root taken and the imaginary unit introduced, we're ready to isolate x and find the final solutions to our equation.
Step 5: Isolating x and Finding the Solutions
To isolate x, we need to add 3 to both sides of the equation: x - 3 = ±2i. This will get x all by itself on one side, revealing the solutions.
Adding 3 to both sides, we get:
x = 3 ± 2i
This tells us that we have two solutions:
x = 3 + 2ix = 3 - 2i
These are complex solutions, meaning they have both a real part (3) and an imaginary part (2i or -2i). This is a perfectly valid outcome, and it demonstrates the versatility of quadratic equations and their solutions. This final step is the culmination of our efforts. By isolating x, we've successfully unveiled the solutions to the equation. The complex solutions we've obtained highlight the richness and complexity of the number system. These solutions may not be as intuitive as real numbers, but they are just as valid and play a crucial role in many areas of mathematics and physics. This step is not just about finding the answers; it's about understanding the nature of those answers and appreciating the broader context in which they exist. So, with x beautifully isolated and the complex solutions in hand, we've successfully completed our mathematical journey and solved the quadratic equation. This is a testament to the power of the completing the square method and the elegance of mathematical reasoning.
Conclusion
So, there you have it! We've successfully solved the quadratic equation 8x² - 48x = -104 by completing the square. We navigated through the steps, made 'a' equal to 1, added the magic constant, factored the perfect square trinomial, and dealt with imaginary numbers like pros. Remember, completing the square is a powerful technique for solving quadratic equations, and with a little practice, you'll be mastering it in no time. Keep practicing, and happy problem-solving!