Solving Square Root Equations: A Step-by-Step Guide

Hey guys! Solving equations with square roots might seem tricky at first, but don't worry, we're going to break it down step by step. We'll use the example equation $x + 4 = \sqrt{2x + 7}$ to guide us through the process. Let's get started!

Understanding the Basics of Square Root Equations

Before we dive into the step-by-step solution, let's understand what square root equations are and why we need a specific approach to solve them. Square root equations are algebraic equations where the variable is found inside a square root. Our main goal is to isolate the variable, but dealing with the square root requires a few extra steps compared to regular equations. To effectively tackle these equations, you will need to square both sides, which will eliminate the square root and allow us to solve for the variable. However, there's a catch! Squaring both sides can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. Therefore, it's super important to check our solutions at the end to make sure they actually work. When you square both sides, you need to carefully consider how that operation impacts the original constraints of the equation. For example, the expression inside a square root must be non-negative. So, when you square both sides, you're essentially changing the domain of the equation. Understanding these nuances helps us avoid pitfalls and ensures we arrive at the correct solution. Basically, we need to verify that our potential solutions are valid by plugging them back into the original equation. This step is critical because it helps us weed out any extraneous solutions that may have popped up during the squaring process. By doing this, we can have confidence that our final answer is correct and that we haven't introduced any errors along the way. Remember, accuracy is key when solving square root equations, so take your time and double-check your work!

Step-by-Step Solution

Okay, let's solve the equation $x + 4 = \sqrt{2x + 7}$ step by step. By following these steps, you'll be able to solve the equation effectively and accurately.

Step 1: Square Both Sides

Our initial equation is $x + 4 = \sqrt{2x + 7}$. To eliminate the square root, we need to square both sides of the equation. Squaring both sides gives us $(x + 4)^2 = (\sqrt{2x + 7})^2$. This is represented by step E. Squaring both sides of an equation is a fundamental algebraic technique used to eliminate square roots and other radicals. When we square both sides, we're applying the same operation to both expressions, maintaining the equality. However, it's crucial to remember that this operation can sometimes introduce extraneous solutions, which are solutions that satisfy the transformed equation but not the original equation. By squaring both sides, we aim to simplify the equation and make it easier to solve for the variable. In this particular case, squaring both sides eliminates the square root, allowing us to proceed with algebraic manipulations to isolate the variable. It's important to be careful when squaring binomials like $(x + 4)^2$, ensuring that we correctly apply the distributive property or use the appropriate formula. Accuracy in this step is essential to avoid errors that can propagate through the rest of the solution. So, take your time and double-check your work to ensure that you've squared both sides correctly.

Step 2: Expand the Left Side

Expanding $(x + 4)^2$ gives us $x^2 + 8x + 16$. So now we have $x^2 + 8x + 16 = 2x + 7$. This corresponds to step C. Expanding the left side involves applying the distributive property or using the formula for squaring a binomial, which is $(a + b)^2 = a^2 + 2ab + b^2$. In this case, $(x + 4)^2$ expands to $x^2 + 2(x)(4) + 4^2$, which simplifies to $x^2 + 8x + 16$. This step is essential because it transforms the equation into a more manageable form, allowing us to combine like terms and rearrange the equation to solve for the variable. Accuracy in expanding the binomial is crucial to avoid errors that can lead to incorrect solutions. So, take your time and carefully apply the distributive property or the binomial formula to ensure that you've expanded the left side correctly. Once the left side is expanded, the equation becomes a quadratic equation, which can be solved using various methods such as factoring, completing the square, or using the quadratic formula.

Step 3: Rearrange the Equation

To solve the quadratic equation, we need to set it to zero. Subtracting $2x$ and $7$ from both sides of $x^2 + 8x + 16 = 2x + 7$ gives us $x^2 + 6x + 9 = 0$. This is step D. Rearranging the equation involves moving all the terms to one side, leaving zero on the other side. This is typically done to set up the equation for factoring or for using the quadratic formula. In this case, we subtract $2x$ and $7$ from both sides of the equation to obtain $x^2 + 6x + 9 = 0$. This step is crucial because it transforms the equation into a standard quadratic form, which is $ax^2 + bx + c = 0$. Once the equation is in this form, we can easily identify the coefficients $a$, $b$, and $c$, which are needed for factoring or for using the quadratic formula. Accuracy in rearranging the equation is essential to avoid errors that can lead to incorrect solutions. So, take your time and carefully move the terms to one side, ensuring that you've changed the signs correctly when necessary. Once the equation is rearranged, we can proceed with factoring or using the quadratic formula to solve for the variable.

Step 4: Factor the Quadratic

The quadratic equation $x^2 + 6x + 9 = 0$ can be factored as $(x + 3)(x + 3) = 0$, which simplifies to $(x + 3)^2 = 0$. This corresponds to step B. Factoring the quadratic equation involves expressing it as a product of two binomials. In this case, $x^2 + 6x + 9$ can be factored as $(x + 3)(x + 3)$, which is equivalent to $(x + 3)^2$. This step is crucial because it allows us to easily identify the solutions to the equation. When a quadratic equation is factored, we can set each factor equal to zero and solve for the variable. Accuracy in factoring the quadratic equation is essential to avoid errors that can lead to incorrect solutions. So, take your time and carefully factor the equation, ensuring that you've found the correct factors. Once the equation is factored, we can proceed with setting each factor equal to zero and solving for the variable.

Step 5: Solve for x

Taking the square root of both sides of $(x + 3)^2 = 0$ gives us $x + 3 = 0$. This is step A. Solving for $x$ involves isolating the variable on one side of the equation. In this case, we subtract 3 from both sides of the equation to obtain $x = -3$. This step is crucial because it gives us the value(s) of $x$ that satisfy the equation. Accuracy in solving for $x$ is essential to avoid errors that can lead to incorrect solutions. So, take your time and carefully isolate the variable, ensuring that you've performed the correct algebraic operations. Once we have a candidate solution, we need to check if that solution is valid.

Step 6: State the Solution

From $x + 3 = 0$, we find that $x = -3$. This corresponds to step F. Therefore, the solution to the equation is $x = -3$. So, the solution to the equation $x+4 = \sqrt{2x+7}$ is $x=-3$.

The Correct Order

So, the correct order of steps is E, C, D, B, A, F.

Verification (Important!)

Let's check if $x = -3$ is a valid solution by plugging it back into the original equation: $x + 4 = \sqrt{2x + 7}$. Substituting $x = -3$, we get $-3 + 4 = \sqrt{2(-3) + 7}$, which simplifies to $1 = \sqrt{-6 + 7}$, and further simplifies to $1 = \sqrt{1}$. Since $1 = 1$, the solution $x = -3$ is valid.

Conclusion

And there you have it! We successfully solved the square root equation $x + 4 = \sqrt{2x + 7}$ by following a step-by-step approach. Remember, the key is to square both sides, rearrange the equation, factor (if possible), solve for $x$, and always verify your solution. Keep practicing, and you'll become a pro at solving these types of equations in no time! Happy solving, guys!