Solving (x+2)^2 = 49 Step By Step Solutions
In this article, we will delve into the process of solving the quadratic equation (x+2)^2 = 49. This type of equation often appears in algebra and is a fundamental concept in mathematics. We will explore different methods to find the solution(s) for x and verify the answers by substitution. Understanding how to solve quadratic equations is crucial for various mathematical applications, including physics, engineering, and computer science. Let’s embark on this mathematical journey together!
Before we dive into the solution, it’s essential to understand the nature of quadratic equations. A quadratic equation is a polynomial equation of the second degree. In simpler terms, it's an equation where the highest power of the variable (in this case, x) is 2. The general form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are constants, and a is not equal to 0. However, the equation (x+2)^2 = 49 is presented in a slightly different form, which we will address shortly.
The equation (x+2)^2 = 49 is a perfect square trinomial equation. This means that the left side of the equation is a binomial squared. To solve this equation, we can use several methods, including taking the square root of both sides, expanding the equation and then factoring, or using the quadratic formula. In this article, we will primarily focus on the method of taking the square root of both sides, as it is the most straightforward approach for this particular equation. This method leverages the property that if a^2 = b^2, then a = ±b. This is a crucial concept to grasp as it highlights that there are often two possible solutions when dealing with squared terms.
To solve the equation (x+2)^2 = 49, the most direct approach is to take the square root of both sides. This method simplifies the equation by eliminating the square on the left side. By taking the square root, we introduce both the positive and negative roots of 49, which is crucial for finding all possible solutions. This step is essential because squaring either a positive or a negative number yields a positive result. Therefore, when reversing the operation, we must consider both possibilities. The square root method is particularly efficient for equations in the form of a squared expression equaling a constant, making it a valuable tool in your mathematical arsenal.
First, let's take the square root of both sides of the equation:
√[(x+2)^2] = ±√49
This simplifies to:
x + 2 = ±7
Now, we have two separate equations to solve:
- x + 2 = 7
- x + 2 = -7
Solving the first equation (x + 2 = 7), we subtract 2 from both sides:
x = 7 - 2
x = 5
So, one solution is x = 5. This is a positive solution, and it's important to remember that quadratic equations often have more than one solution due to the squared term.
Next, let's solve the second equation (x + 2 = -7). Again, we subtract 2 from both sides:
x = -7 - 2
x = -9
Thus, the second solution is x = -9. This is a negative solution, highlighting the importance of considering both positive and negative roots when solving quadratic equations. By finding both solutions, we ensure a complete and accurate answer to the problem.
Another way to solve the equation (x+2)^2 = 49 is by expanding the left side, rearranging the equation into the standard quadratic form, and then factoring. This method is slightly more involved than taking the square root directly, but it reinforces important algebraic skills and can be useful for solving other types of quadratic equations. Expanding the equation involves multiplying out the squared term, which transforms the equation into a more familiar quadratic form. Factoring, on the other hand, is the process of breaking down the quadratic expression into two binomial expressions, which then allows us to find the solutions by setting each factor equal to zero.
First, expand the left side of the equation (x+2)^2 = 49:
(x+2)^2 = (x+2)(x+2)
Using the FOIL (First, Outer, Inner, Last) method or the distributive property, we get:
(x+2)(x+2) = x^2 + 2x + 2x + 4
Combine like terms:
x^2 + 4x + 4
Now, substitute this back into the original equation:
x^2 + 4x + 4 = 49
To solve by factoring, we need to set the equation equal to zero. Subtract 49 from both sides:
x^2 + 4x + 4 - 49 = 0
Simplify:
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.
So, we can factor the expression as follows:
(x + 9)(x - 5) = 0
To find the solutions for x, we set each factor equal to zero:
- x + 9 = 0
- x - 5 = 0
Solving the first equation (x + 9 = 0), we subtract 9 from both sides:
x = -9
Solving the second equation (x - 5 = 0), we add 5 to both sides:
x = 5
Thus, the solutions are x = -9 and x = 5. This method provides the same solutions as taking the square root directly, but it demonstrates a different algebraic approach that can be applied to a broader range of quadratic equations.
To ensure the accuracy of our solutions, it's essential to verify them by substituting them back into the original equation (x+2)^2 = 49. This process confirms that the values we found for x indeed satisfy the equation. Verification is a crucial step in problem-solving, as it helps to catch any potential errors made during the solution process. By substituting the values back into the original equation, we are essentially checking if the left-hand side of the equation equals the right-hand side when x is replaced with our solutions.
Let's start by verifying the first solution, x = 5:
Substitute x = 5 into the original equation:
(5 + 2)^2 = 49
Simplify:
(7)^2 = 49
49 = 49
Since the left side equals the right side, x = 5 is a valid solution. This verification confirms that our algebraic manipulations were correct and that the value we found for x is indeed a solution to the equation. Now, let's move on to verifying the second solution.
Next, let's verify the second solution, x = -9:
Substitute x = -9 into the original equation:
(-9 + 2)^2 = 49
Simplify:
(-7)^2 = 49
49 = 49
Since the left side equals the right side, x = -9 is also a valid solution. This further reinforces the importance of considering both positive and negative roots when solving quadratic equations. By verifying both solutions, we can confidently state that we have found all possible solutions to the given equation.
In conclusion, the solutions to the equation (x+2)^2 = 49 are x = 5 and x = -9. We arrived at these solutions using two different methods: taking the square root of both sides and expanding and factoring. Both methods yielded the same results, demonstrating the versatility of algebraic techniques in solving quadratic equations. We also verified our solutions by substituting them back into the original equation, ensuring their accuracy. Understanding how to solve quadratic equations is a fundamental skill in mathematics, and mastering these techniques will be beneficial in various mathematical and real-world applications. Remember to always consider all possible solutions and verify your answers to ensure accuracy. Happy solving!
Based on our calculations and verifications, the correct options are:
- B. x = 5
- D. x = -9
