Solve The Equation (x+2)^2=49. Which Of The Following Are Solutions? A. X=5 B. X=-7 C. X=-5 D. X=7 E. X=-9

This article provides a comprehensive guide to solving the equation $(x+2)^2 = 49$. We will explore different methods to find the solution(s) and verify them. This is a fundamental problem in algebra, often encountered in introductory courses and serves as a building block for more advanced topics. Understanding how to solve such equations is crucial for success in mathematics. We will delve into the step-by-step process, ensuring clarity and a thorough understanding of the underlying principles. Let's embark on this mathematical journey together!

Understanding the Problem

Before diving into the solution, it's crucial to understand the problem we are tackling. The equation $(x+2)^2 = 49$ is a quadratic equation in disguise. It involves a variable, x, and the highest power of x is 2 when the equation is expanded. However, the equation is presented in a factored form, which simplifies the solution process. Our goal is to find the value(s) of x that make the equation true. In other words, we are looking for the number(s) that, when added to 2 and squared, result in 49. The presence of the square suggests that there might be two possible solutions, as both a positive and a negative number, when squared, can yield a positive result. Recognizing these key aspects is essential for choosing the most efficient solution strategy and avoiding common pitfalls. We will explore two primary methods for solving this equation: the square root method and the expansion and factoring method. Each method offers a unique perspective on the problem and provides valuable insights into solving quadratic equations.

Method 1: The Square Root Method

The square root method is a direct and efficient approach for solving equations of the form $(ax + b)^2 = c$, where a, b, and c are constants. This method leverages the inverse relationship between squaring and taking the square root. The core idea is to isolate the squared term and then take the square root of both sides of the equation. This introduces both positive and negative roots, leading to two potential solutions. In our specific case, the equation is $(x+2)^2 = 49$. The squared term is already isolated, so our first step is to take the square root of both sides. Remember that when we take the square root, we must consider both the positive and negative roots of 49, which are 7 and -7 respectively. This gives us two separate equations: $x + 2 = 7$ and $x + 2 = -7$. Now, we solve each equation for x. For the first equation, $x + 2 = 7$, we subtract 2 from both sides to get $x = 5$. For the second equation, $x + 2 = -7$, we subtract 2 from both sides to get $x = -9$. Therefore, the solutions obtained using the square root method are $x = 5$ and $x = -9$. This method highlights the importance of considering both positive and negative roots when dealing with squared terms. Let's now explore the second method, which involves expanding and factoring the equation.

Method 2: Expansion and Factoring

The alternative approach involves expanding and factoring. This method is more general and can be applied to a wider range of quadratic equations. It involves expanding the squared term, rearranging the equation into standard quadratic form, and then factoring the resulting quadratic expression. In our case, the equation is $(x+2)^2 = 49$. First, we expand the left side of the equation. Using the formula $(a + b)^2 = a^2 + 2ab + b^2$, we get $x^2 + 4x + 4 = 49$. Next, we want to rewrite the equation in the standard quadratic form, which is $ax^2 + bx + c = 0$. To do this, we subtract 49 from both sides of the equation, resulting in $x^2 + 4x + 4 - 49 = 0$, which simplifies to $x^2 + 4x - 45 = 0$. Now, we need to factor the quadratic expression $x^2 + 4x - 45$. We are looking for two numbers that multiply to -45 and add up to 4. These numbers are 9 and -5. Thus, we can factor the quadratic expression as $(x + 9)(x - 5) = 0$. According to the zero-product property, if the product of two factors is zero, then at least one of the factors must be zero. Therefore, either $x + 9 = 0$ or $x - 5 = 0$. Solving these two equations, we get $x = -9$ and $x = 5$. These solutions are consistent with those obtained using the square root method. This method emphasizes the importance of understanding quadratic forms and factoring techniques.

Verification of Solutions

Verification is a crucial step in the problem-solving process. It ensures that the solutions we have obtained are correct and satisfy the original equation. To verify our solutions, we substitute each value of x back into the original equation $(x+2)^2 = 49$ and check if the equation holds true. Let's start with $x = 5$. Substituting this value into the equation, we get $(5 + 2)^2 = 49$, which simplifies to $7^2 = 49$, which is indeed true. Therefore, $x = 5$ is a valid solution. Now, let's verify the second solution, $x = -9$. Substituting this value into the equation, we get $(-9 + 2)^2 = 49$, which simplifies to $(-7)^2 = 49$, which is also true. Therefore, $x = -9$ is also a valid solution. Since both solutions satisfy the original equation, we can confidently conclude that they are correct. Verification not only confirms the correctness of the solutions but also enhances our understanding of the problem and the solution process. It's a habit that should be cultivated in mathematics and problem-solving in general. In this case, we have successfully verified that both $x = 5$ and $x = -9$ are the solutions to the equation $(x+2)^2 = 49$.

Selecting the Correct Options

Now that we have found the solutions to the equation $(x+2)^2 = 49$, which are $x = 5$ and $x = -9$, we can identify the correct options from the given choices. The options provided are:

A. $x=5$ B. $x=-7$ C. $x=-5$ D. $x=7$ E. $x=-9$

Comparing our solutions with the options, we can see that options A and E match our solutions. Option A states $x = 5$, which is one of our solutions. Option E states $x = -9$, which is the other solution we found. The other options, B, C, and D, do not match our solutions. Option B states $x = -7$, Option C states $x = -5$, and Option D states $x = 7$. None of these values satisfy the original equation when substituted. Therefore, the correct options are A and E. This step highlights the importance of carefully comparing the solutions obtained with the given options to ensure accurate selection. It's a crucial final step in the problem-solving process that prevents errors and ensures that the correct answers are chosen.

Conclusion

In this article, we have successfully solved the equation $(x+2)^2 = 49$ using two different methods: the square root method and the expansion and factoring method. Both methods led us to the same solutions, $x = 5$ and $x = -9$. We also verified these solutions by substituting them back into the original equation, confirming their correctness. Finally, we identified the correct options from the given choices, which are A ($x = 5$) and E ($x = -9$). This problem serves as a valuable example of how to solve quadratic equations in a factored form. The square root method provides a direct approach for equations of this type, while the expansion and factoring method is a more general technique applicable to a wider range of quadratic equations. Understanding both methods enhances problem-solving skills and provides a deeper understanding of algebraic concepts. Furthermore, the importance of verification cannot be overstated. It is a crucial step that ensures the accuracy of the solutions and builds confidence in the problem-solving process. By mastering these techniques, you will be well-equipped to tackle more complex mathematical problems in the future. Remember to always approach problems systematically, understand the underlying principles, and verify your solutions to ensure accuracy.