Solving 2x² + 8x = X² - 16: A Step-by-Step Guide
Hey there, math enthusiasts! Ever stumbled upon a quadratic equation that seemed like a tangled mess? Well, fret no more! In this article, we're going to break down the equation 2x² + 8x = x² - 16 step by step, making sure you not only understand the solution but also the why behind it. Think of this as your friendly guide to conquering quadratic equations, turning what might seem daunting into a walk in the park. So, grab your pencils, and let's dive in!
Understanding the Quadratic Equation
Before we jump into solving, let's get cozy with what we're dealing with. Quadratic equations are those cool cats that have a variable raised to the power of two – that x² we see in our equation. They often pop up in real-world scenarios, from figuring out the trajectory of a ball to designing the perfect curve in architecture. The general form of a quadratic equation is ax² + bx + c = 0, where 'a', 'b', and 'c' are constants. Our mission? To find the values of 'x' that make the equation true. These values are also known as the roots or solutions of the equation. Now, why is this important? Well, understanding the roots helps us understand the behavior of the quadratic function, which can be visualized as a parabola on a graph. The roots are the points where the parabola intersects the x-axis. So, when we solve for 'x', we're essentially finding these crucial intersection points. But enough with the theory, let's get practical and see how we can transform our equation into the standard form. We need to rearrange 2x² + 8x = x² - 16 so that it looks like ax² + bx + c = 0. This involves moving all terms to one side of the equation. Think of it as tidying up a messy room – we want everything in its place! By subtracting x² from both sides, we get x² + 8x = -16. Then, adding 16 to both sides gives us x² + 8x + 16 = 0. Voila! Our equation is now in the standard form, ready for us to work our magic.
Step-by-Step Solution
Now that our equation is in the standard form x² + 8x + 16 = 0, it's time to roll up our sleeves and find those elusive solutions. There are several methods we can use, but today, we're going to focus on factoring – a technique that's like detective work for numbers. Factoring involves breaking down the quadratic expression into two binomials (expressions with two terms) that, when multiplied together, give us the original equation. Think of it as reverse multiplication. We need to find two numbers that add up to the coefficient of our 'x' term (which is 8 in this case) and multiply to the constant term (which is 16). Let's put on our thinking caps. What two numbers fit the bill? If you guessed 4 and 4, you're spot on! Because 4 + 4 = 8 and 4 * 4 = 16. So, we can rewrite our equation as (x + 4)(x + 4) = 0, or more simply, (x + 4)² = 0. See how we transformed a seemingly complex equation into something much more manageable? Now comes the crucial step: the Zero Product Property. This property states that if the product of two factors is zero, then at least one of the factors must be zero. In our case, we have (x + 4) multiplied by itself equaling zero. This means that (x + 4) must be zero. So, we set x + 4 = 0 and solve for 'x'. Subtracting 4 from both sides, we get x = -4. And there you have it! The solution to our quadratic equation is x = -4. But wait, there's a little more to the story. Notice that we have (x + 4) appearing twice in our factored form. This means that x = -4 is a repeated root. What does this mean geometrically? It means that the parabola represented by our quadratic equation touches the x-axis at only one point, x = -4. It doesn't cross the axis, but rather just kisses it. This gives us a deeper understanding of the behavior of the quadratic function.
Verification and Alternative Methods
We've found our solution, x = -4, but like any good mathematician, we should always verify our answer. It's like double-checking your work before submitting a test – ensures you haven't made any silly mistakes. To verify, we substitute x = -4 back into our original equation, 2x² + 8x = x² - 16, and see if both sides of the equation are equal. Plugging in x = -4, we get 2(-4)² + 8(-4) = (-4)² - 16. Simplifying, we have 2(16) - 32 = 16 - 16, which becomes 32 - 32 = 0. And guess what? 0 = 0! Our solution checks out. We can breathe a sigh of relief knowing we've cracked the code. Now, while factoring was our chosen method today, it's not the only tool in the quadratic equation toolbox. There are other methods, such as the quadratic formula and completing the square, that can be used to solve these types of equations. The quadratic formula, in particular, is a powerful tool that can solve any quadratic equation, regardless of whether it can be easily factored. It's like the Swiss Army knife of quadratic equations! The formula is given by:
x = (-b ± √(b² - 4ac)) / 2a
Where 'a', 'b', and 'c' are the coefficients from our standard form equation, ax² + bx + c = 0. Applying this formula to our equation x² + 8x + 16 = 0, where a = 1, b = 8, and c = 16, we get:
x = (-8 ± √(8² - 4 * 1 * 16)) / (2 * 1)
Simplifying, we have:
x = (-8 ± √(64 - 64)) / 2
x = (-8 ± √0) / 2
x = -8 / 2
x = -4
As you can see, the quadratic formula gives us the same solution, x = -4, confirming our earlier result. This highlights the beauty of mathematics – different paths can lead to the same destination. Each method has its strengths and weaknesses, and choosing the right one often depends on the specific equation you're dealing with. Factoring is great when the equation can be easily factored, while the quadratic formula is a reliable workhorse that always gets the job done. So, feel free to experiment with different methods and find the ones that resonate with you.
Real-World Applications
Okay, we've conquered the equation 2x² + 8x = x² - 16, but you might be wondering,
