Solving $2x^2 + 8x = X^2 - 16$: A Step-by-Step Guide

Let's dive into solving the quadratic equation $2x^2 + 8x = x^2 - 16$. Quadratic equations might seem intimidating at first, but with a step-by-step approach, they become much easier to handle. In this guide, we'll break down the process, making it simple and clear. We'll transform the equation into its standard form, identify the coefficients, and then use these coefficients to find the solutions. By the end of this article, you’ll not only know how to solve this specific equation but also understand the general method for tackling quadratic equations. Understanding these steps is essential for anyone studying algebra, calculus, or any related field in mathematics. Once you grasp the fundamentals, you'll be able to apply this knowledge to more complex problems and real-world scenarios. The beauty of mathematics lies in its logical progression, and mastering quadratic equations is a significant step in building a strong foundation.

Step 1: Rearrange the Equation

Our first step is to rearrange the given equation $2x^2 + 8x = x^2 - 16$ so that it's in the standard quadratic form, which is $ax^2 + bx + c = 0$. This form makes it easier to identify the coefficients we'll need later.

To do this, we'll subtract $x^2$ from both sides of the equation:

$2x^2 - x^2 + 8x = x^2 - x^2 - 16$

This simplifies to:

$x^2 + 8x = -16$

Next, we want to move the constant term to the left side of the equation. We'll add 16 to both sides:

$x^2 + 8x + 16 = -16 + 16$

Which simplifies to:

$x^2 + 8x + 16 = 0$

Now, our equation is in the standard quadratic form: $ax^2 + bx + c = 0$, where $a = 1$, $b = 8$, and $c = 16$. Getting the equation into this standard form is crucial because it sets the stage for applying various solution methods, such as factoring, completing the square, or using the quadratic formula. By rearranging the equation, we've made it easier to identify the key components that we need to solve it. This methodical approach ensures that we don't miss any steps and that we can accurately find the solutions to the equation. Without this initial rearrangement, the subsequent steps would be much more difficult and prone to error. So, take your time with this step and ensure that you've correctly transformed the equation into the standard quadratic form.

Step 2: Factor the Quadratic Expression

Now that we have the quadratic equation in the standard form $x^2 + 8x + 16 = 0$, we can try to factor the quadratic expression. Factoring involves finding two binomials that, when multiplied together, give us the original quadratic expression. In other words, we are looking for two expressions of the form $(x + m)(x + n)$ such that their product equals $x^2 + 8x + 16$.

We need to find two numbers, $m$ and $n$, that satisfy two conditions:

1. Their product ($m ) times $n$) equals the constant term, which is 16.
2. Their sum ($m + n$) equals the coefficient of the $x$ term, which is 8.

Let's think about factors of 16:

• 1 and 16 (sum is 17)
• 2 and 8 (sum is 10)
• 4 and 4 (sum is 8)

Ah, we found it! The numbers 4 and 4 satisfy both conditions. So, we can write the quadratic expression as:

$x^2 + 8x + 16 = (x + 4)(x + 4)$

Or, more compactly:

$x^2 + 8x + 16 = (x + 4)^2$

Therefore, our equation becomes:

$(x + 4)^2 = 0$

Factoring the quadratic expression simplifies the equation and makes it easier to find the solutions. It's like breaking down a complex problem into smaller, more manageable parts. When you can factor a quadratic expression, you're essentially rewriting it in a form that directly reveals its roots. In this case, we found that $x^2 + 8x + 16$ can be written as $(x + 4)^2$, which tells us that the equation has a repeated root. Understanding the concept of factoring is fundamental in algebra and is a powerful tool for solving quadratic equations and other polynomial equations. Not all quadratic expressions can be easily factored, but when they can, it's often the quickest and most straightforward way to find the solutions.

Step 3: Solve for x

Now that we have factored the equation as $(x + 4)^2 = 0$, solving for $x$ becomes much simpler. Since $(x + 4)^2 = 0$, it must be the case that $x + 4 = 0$. This is because the only number that, when squared, equals zero is zero itself.

So, we have:

$x + 4 = 0$

To isolate $x$, we subtract 4 from both sides of the equation:

$x + 4 - 4 = 0 - 4$

This simplifies to:

$x = -4$

Thus, the solution to the quadratic equation $2x^2 + 8x = x^2 - 16$ is $x = -4$. Since the factor $(x + 4)$ appears twice in the factored form $(x + 4)^2$, we say that $x = -4$ is a repeated root or a root with multiplicity 2. This means that the quadratic equation has only one distinct solution.

Solving for $x$ is the ultimate goal of the entire process. It's the step where we finally find the value(s) of $x$ that satisfy the original equation. In this case, we found that $x = -4$ is the only solution. This means that if we substitute $-4$ for $x$ in the original equation, the equation will hold true. Always remember to check your solutions by substituting them back into the original equation to ensure that they are correct. The ability to solve for x is a crucial skill in mathematics and is used extensively in various fields, including physics, engineering, and economics. Mastering this skill will enable you to tackle a wide range of problems and make accurate predictions.

Conclusion

In summary, to solve the quadratic equation $2x^2 + 8x = x^2 - 16$, we followed these steps:

1. Rearranged the equation into the standard form: $x^2 + 8x + 16 = 0$.
2. Factored the quadratic expression: $(x + 4)^2 = 0$.
3. Solved for $x$: $x = -4$.

Therefore, the solution to the equation is $x = -4$. This comprehensive guide has shown you how to solve this specific quadratic equation and has also provided you with a general approach for tackling similar equations. Remember to practice these steps with different quadratic equations to build your confidence and skills. Understanding quadratic equations is fundamental to grasping more advanced mathematical concepts, and mastering them will undoubtedly benefit you in your future studies and endeavors.

For more information on quadratic equations, visit Khan Academy's Quadratic Equations Section.