Solving Radical Equations: A Step-by-Step Guide

Welcome, math enthusiasts! Today, we're diving into the world of radical equations and, more specifically, how to solve for x in the equation ${\sqrt{x+16} = x-4}$. Don't worry if the idea of radicals seems intimidating; we'll break it down into manageable steps. This guide is designed to be clear, concise, and easy to follow, whether you're a seasoned mathlete or just starting out. Our aim is to not only find the solution but also to understand the why behind each step. Let's get started!

Understanding the Basics: Radical Equations

Before we jump into the equation, let's quickly review what a radical equation is. Simply put, a radical equation is an equation that contains a radical expression, such as a square root, cube root, or any other root. Our equation, ${\sqrt{x+16} = x-4}$, is a radical equation because it has a square root. The key to solving these equations is to isolate the radical and then eliminate it by using the inverse operation – raising both sides of the equation to the power that matches the root's index (for a square root, we square; for a cube root, we cube, and so on). This process can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original one. Therefore, always check your solutions in the original equation. In our case, we have a square root, so our inverse operation will be squaring. This fundamental understanding is crucial for tackling more complex problems. Remember that the goal is to unravel the equation systematically, ensuring that each manipulation maintains the equation's balance. Always be mindful of the potential for extraneous solutions, and always verify your answers! It's like double-checking your work in any other field – it ensures accuracy and prevents errors.

Now, let's explore this approach in solving equations. The primary purpose of solving a radical equation is to determine the values of the variable(s) that satisfy the equation. The process begins with isolating the radical term on one side of the equation. This isolation simplifies the equation, allowing for easier manipulation. Once the radical term is isolated, the next step involves eliminating the radical. This is accomplished by raising both sides of the equation to the power that corresponds to the index of the radical. For example, if the equation involves a square root (index 2), both sides are squared. If the equation involves a cube root (index 3), both sides are cubed. This process is crucial because it removes the radical, transforming the equation into a more familiar form (typically a linear or quadratic equation). After removing the radical, the resulting equation is solved using standard algebraic techniques. This may involve simplifying, factoring, or using the quadratic formula, depending on the nature of the equation. However, the solutions obtained from the transformed equation must always be checked in the original equation. This step is critical because the process of eliminating the radical can introduce extraneous solutions. Extraneous solutions are values that satisfy the transformed equation but do not satisfy the original radical equation. Checking the solutions involves substituting each value back into the original equation to verify that it holds true. If a solution does not satisfy the original equation, it is discarded as extraneous. This verification step ensures that only valid solutions are accepted, maintaining the integrity and accuracy of the solution. By following these steps, you can effectively solve radical equations, ensuring that the found solutions are both correct and applicable to the initial equation. Remember, each step plays a crucial role in the process, guaranteeing a logical and accurate outcome. Let's begin the practical part with our equation.

Step-by-Step Solution to ${\sqrt{x+16} = x-4}$

Let's meticulously solve the equation ${\sqrt{x+16} = x-4}$. We'll break it down into easy-to-follow steps.

Step 1: Isolate the Radical. In this case, the radical ${\sqrt{x+16}}$ is already isolated on the left side of the equation. This simplifies our initial steps considerably!

Step 2: Square Both Sides. To eliminate the square root, we square both sides of the equation. This gives us:

${(\sqrt{x+16})^2 = (x-4)^2}$

Simplifying this, we get:

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

Notice that the square root is gone, and we now have a quadratic equation. This is a very common transformation when dealing with radical equations. The key here is to apply the squaring operation carefully to both sides, ensuring that the entire expression on each side is squared correctly. Pay close attention to the order of operations and the correct application of algebraic rules during this phase.

Step 3: Simplify and Rearrange. Now, let's move all terms to one side to set the equation to zero:

${0 = x^2 - 9x}$

This gives us a simpler quadratic equation to work with.

Step 4: Solve the Quadratic Equation. We can factor out an x from the quadratic equation:

${x(x - 9) = 0}$

This yields two possible solutions: ${x = 0}$ and ${x = 9}$. Factoring the equation is a common approach to solving quadratic equations, especially when the equation is relatively simple. The factored form allows us to quickly identify the values of x that make the equation true. Setting each factor equal to zero is a critical step, as it helps identify all possible solutions to the equation. Keep in mind that not all quadratic equations can be factored easily, and in such cases, other methods like the quadratic formula might be necessary. But in our case, factoring provided us with two potential solutions, which we will now check for validity.

Step 5: Check for Extraneous Solutions. This is a crucial step. We must substitute our solutions back into the original equation ${\sqrt{x+16} = x-4}$ to see if they work.

• Check ${x = 0}$: ${\sqrt{0+16} = 0-4}$ ${\sqrt{16} = -4}$ ${4 = -4}$ This is false. Therefore, ${x = 0}$ is an extraneous solution.

• Check ${x = 9}$: ${\sqrt{9+16} = 9-4}$ ${\sqrt{25} = 5}$ ${5 = 5}$ This is true. Therefore, ${x = 9}$ is a valid solution.

Step 6: State the Solution. The only valid solution to the equation ${\sqrt{x+16} = x-4}$ is ${x = 9}$.

Why Checking Solutions Matters

As we saw, checking solutions is critical because squaring both sides of an equation can introduce extraneous solutions. Extraneous solutions arise because the squaring operation can