What Are The Solutions To The Quadratic Equation (x+3)^2 = 49? Provide A Step-by-step Explanation Of How To Solve The Equation.

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In the realm of mathematics, quadratic equations hold a significant position, often appearing in various scientific and engineering applications. A quadratic equation is a polynomial equation of the second degree, meaning the highest power of the variable is 2. Solving these equations involves finding the values of the variable that satisfy the equation, also known as the roots or solutions. In this article, we delve into the intricacies of solving the quadratic equation (x+3)^2 = 49, exploring different methods and arriving at the correct solutions.

Understanding Quadratic Equations

Before diving into the specifics of solving (x+3)^2 = 49, it's crucial to grasp the fundamental concepts of quadratic equations. A quadratic equation generally takes the form ax^2 + bx + c = 0, where a, b, and c are constants, and x represents the variable. The solutions to a quadratic equation can be real or complex numbers, and there are several methods to find them, including factoring, completing the square, and using the quadratic formula.

Methods for Solving Quadratic Equations

  1. Factoring: This method involves expressing the quadratic equation as a product of two linear factors. If the equation can be factored, the solutions can be easily found by setting each factor equal to zero and solving for x. However, not all quadratic equations can be factored easily.

  2. Completing the Square: This method involves manipulating the equation algebraically to create a perfect square trinomial on one side. This allows us to take the square root of both sides and solve for x. Completing the square is a versatile method that works for any quadratic equation.

  3. Quadratic Formula: The quadratic formula is a general formula that provides the solutions to any quadratic equation. It is given by x = (-b ± √(b^2 - 4ac)) / 2a. This formula is particularly useful when factoring or completing the square is difficult or time-consuming.

Solving (x+3)^2 = 49: A Step-by-Step Approach

Now, let's focus on solving the given quadratic equation (x+3)^2 = 49. We will explore two primary methods: the square root property and expansion and factoring.

Method 1: Square Root Property

The square root property is a direct approach when the quadratic equation is in a form where a squared term is isolated on one side. In this case, we have (x+3)^2 = 49. To apply the square root property, we take the square root of both sides of the equation:

√((x+3)^2) = ±√49

This simplifies to:

x + 3 = ±7

Now, we have two separate equations to solve:

  1. x + 3 = 7
  2. x + 3 = -7

Solving the first equation, we subtract 3 from both sides:

x = 7 - 3

x = 4

Solving the second equation, we also subtract 3 from both sides:

x = -7 - 3

x = -10

Therefore, the solutions obtained using the square root property are x = 4 and x = -10. This method is efficient and straightforward for equations in this specific form.

Method 2: Expansion and Factoring

Another approach to solve (x+3)^2 = 49 involves expanding the squared term and then rearranging the equation into the standard quadratic form ax^2 + bx + c = 0. Let's proceed with this method:

First, we expand (x+3)^2:

(x+3)^2 = (x+3)(x+3) = x^2 + 3x + 3x + 9 = x^2 + 6x + 9

Now, we substitute this back into the original equation:

x^2 + 6x + 9 = 49

Next, we subtract 49 from both sides to set the equation to zero:

x^2 + 6x + 9 - 49 = 0

x^2 + 6x - 40 = 0

Now we have a quadratic equation in the standard form. To solve this, we attempt to factor the quadratic expression. We are looking for two numbers that multiply to -40 and add up to 6. These numbers are 10 and -4.

Thus, we can factor the quadratic expression as follows:

(x + 10)(x - 4) = 0

Setting each factor equal to zero gives us the solutions:

  1. x + 10 = 0 x = -10

  2. x - 4 = 0 x = 4

Therefore, the solutions obtained by expansion and factoring are also x = 4 and x = -10. This method provides a more general approach that can be applied to a wider range of quadratic equations.

Comparing the Methods

Both the square root property and expansion and factoring methods lead to the same solutions for the equation (x+3)^2 = 49. The square root property is quicker and more direct when the equation is in the form (x+k)^2 = c, while expansion and factoring is a more general approach that can be used for any quadratic equation. Understanding both methods enhances your problem-solving toolkit and allows you to choose the most efficient method for a given equation.

Verifying the Solutions

To ensure the accuracy of our solutions, it's essential to verify them by substituting them back into the original equation. Let's verify x = 4 and x = -10 in the equation (x+3)^2 = 49.

Verification for x = 4

Substituting x = 4 into the equation:

(4 + 3)^2 = 49

(7)^2 = 49

49 = 49

The equation holds true, confirming that x = 4 is a valid solution.

Verification for x = -10

Substituting x = -10 into the equation:

(-10 + 3)^2 = 49

(-7)^2 = 49

49 = 49

The equation also holds true, confirming that x = -10 is a valid solution.

Conclusion

In conclusion, the solutions to the quadratic equation (x+3)^2 = 49 are x = 4 and x = -10. We arrived at these solutions using two distinct methods: the square root property and expansion and factoring. Both methods demonstrate different problem-solving approaches, and understanding them broadens your mathematical skills. Additionally, we verified the solutions by substituting them back into the original equation, ensuring their accuracy. Mastering the techniques for solving quadratic equations is crucial for success in various mathematical and scientific domains.

The correct answer is C. x=4 and x=-10.