Solving $8x^2 - 48x = -104$: A Step-by-Step Guide
Hey guys! Today, we're diving into the exciting world of quadratic equations. Specifically, we're going to tackle the equation $8x^2 - 48x = -104$. Our main goal is to transform this equation into a more manageable form, where the coefficient of the $x^2$ term (which we often call 'a') is equal to 1. This makes it much easier to solve using methods like completing the square or the quadratic formula. So, let's break it down step by step and make sure we understand each move we make. Get ready to sharpen your pencils and put on your math hats – it's going to be a fun ride!
Step 1: Divide the entire equation by 8
Our initial equation is $8x^2 - 48x = -104$. To get the coefficient of $x^2$ to be 1, we need to divide every term in the equation by 8. This is a crucial first step because it simplifies the equation and sets us up for easier manipulation later on. So, let's do the division:
This simplifies to:
Now, we have a quadratic equation where the coefficient of $x^2$ is indeed 1. This is exactly what we wanted! Notice how dividing each term equally maintains the balance of the equation. We haven't changed the solutions; we've just reshaped the equation into a more friendly form. This is a common technique in algebra, and you'll find it super useful as you encounter more complex problems. By dividing by 8, we've effectively normalized the equation, making it easier to work with in subsequent steps. Think of it as setting the stage for the main performance – all the key elements are now in place for us to find the solutions.
Step 2: Expressing the Equation in the Desired Form $x^2 + _ x = _$
Now that we've divided the entire equation by 8, we've arrived at a much cleaner form: $x^2 - 6x = -13$. This equation already fits the desired format of $x^2 + _ x = _$, where the coefficient of $x^2$ is 1. In our case, the blank space before the 'x' represents the coefficient -6, and the blank space on the right-hand side represents the constant -13. So, we can directly see that the equation is in the form we were aiming for. This form is particularly useful because it allows us to easily apply techniques like completing the square to solve for x. By having the equation in this format, we've essentially isolated the quadratic and linear terms on one side, making it simpler to manipulate and find the roots. This step is crucial in setting up the equation for further analysis and solution. We've successfully transformed the original equation into a more manageable structure, paving the way for us to find the values of x that satisfy the equation. Think of it as organizing your tools before starting a project – everything is now in its place, ready for the next step.
Step 3: Completing the Square (Optional, but a common next step)
While our original goal was just to rewrite the equation in the form $x^2 + _ x = _$, let's briefly touch upon a common technique used to solve such equations: completing the square. This method is incredibly powerful for finding the solutions (or roots) of a quadratic equation. Our equation currently stands at $x^2 - 6x = -13$. To complete the square, we need to add a specific constant to both sides of the equation. This constant is calculated as the square of half the coefficient of our 'x' term. In this case, the coefficient of 'x' is -6. Half of -6 is -3, and the square of -3 is 9. So, we add 9 to both sides:
The left side now forms a perfect square trinomial, which can be factored as:
Completing the square transforms the quadratic equation into a form where we can easily isolate x. Notice how the left side is now a squared term. This is the key idea behind completing the square – we've created a perfect square that simplifies the equation. The right side simplifies to -4. At this point, we can take the square root of both sides (remembering to consider both positive and negative roots) to solve for x. This method not only helps us find the solutions but also provides valuable insights into the nature of the roots (real or complex). Although we won't fully solve for x here, this step demonstrates how the form $x^2 + _ x = _$ sets the stage for completing the square, a fundamental technique in algebra. Think of it as building a bridge to the solution – we've laid the groundwork, and the path to finding x is now much clearer.
Step 4: Solving for x (Continuing from Completing the Square)
Building upon our previous step of completing the square, we arrived at the equation $(x - 3)^2 = -4$. Now, let's actually solve for x. To do this, we need to take the square root of both sides of the equation. Remember that when taking the square root, we must consider both the positive and negative roots. This gives us:
Since we have the square root of a negative number, we know that the solutions will be complex numbers. Recall that the square root of -1 is denoted by 'i' (the imaginary unit). So, √(-4) can be rewritten as √(4 * -1) = √(4) * √(-1) = 2i. Now our equation looks like this:
To isolate x, we add 3 to both sides:
This gives us two complex solutions: x = 3 + 2i and x = 3 - 2i. These are the values of x that satisfy the original equation $8x^2 - 48x = -104$. Notice how completing the square led us directly to these solutions. By transforming the equation into a squared term, we made it possible to isolate x and find its values, even when they are complex numbers. This process highlights the power of algebraic manipulation in solving equations. We started with a quadratic equation and, through a series of strategic steps, arrived at the solutions. This journey showcases the elegance and effectiveness of mathematical techniques in unraveling complex problems. Think of it as solving a puzzle – each step reveals a new piece, ultimately leading to the complete picture.
Conclusion: Mastering Quadratic Equations
Wow, we've journeyed through the process of transforming and solving a quadratic equation, starting with $8x^2 - 48x = -104$! We successfully rewrote the equation in the form $x^2 + _ x = _$, which is a crucial step for further manipulation. We then explored the technique of completing the square, which allowed us to find the complex solutions for x: 3 + 2i and 3 - 2i. This exercise highlights the importance of understanding algebraic techniques and how they can be applied to solve problems. Mastering quadratic equations is a fundamental skill in algebra and opens doors to more advanced mathematical concepts. By breaking down the problem into manageable steps, we made the process clear and understandable. Each step, from dividing by the coefficient of $x^2$ to completing the square, serves a specific purpose in guiding us towards the solution. This systematic approach is key to tackling any mathematical challenge. Remember, practice makes perfect! The more you work with quadratic equations, the more comfortable and confident you'll become. So, keep exploring, keep solving, and keep enjoying the beauty of mathematics! Think of it as building a strong foundation – each equation you solve adds another brick to your understanding, making you a more skilled and capable mathematician. You've got this!
