Solve The Equation: $2(x+3)^2 - 242 = 0$Which Of The Following Are The Values Of $x$ That Satisfy The Given Equation?A. $x = -8, X = 14$ B. $x = -14, X = 8$ C. $x = -14, X = 14$ D. $x = -8, X = 8$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific quadratic equation, $2(x+3)^2 - 242 = 0$, and explore the different values of $x$ that satisfy the given equation.

Understanding Quadratic Equations

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable. Quadratic equations can be solved using various methods, including factoring, completing the square, and the quadratic formula.

The Given Equation

The given equation is $2(x+3)^2 - 242 = 0$. To solve this equation, we need to isolate the variable $x$. The first step is to add $242$ to both sides of the equation, which gives us:

$$2(x+3)^2 = 242$$

Expanding the Equation

Next, we need to expand the left-hand side of the equation using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, $a = x$ and $b = 3$, so we have:

$$2(x^2 + 6x + 9) = 242$$

Simplifying the Equation

Now, we can simplify the equation by distributing the $2$ to the terms inside the parentheses:

$$2x^2 + 12x + 18 = 242$$

Subtracting 242 from Both Sides

To isolate the variable $x$, we need to subtract $242$ from both sides of the equation:

$$2x^2 + 12x - 224 = 0$$

Dividing Both Sides by 2

Next, we can divide both sides of the equation by $2$ to simplify it further:

$$x^2 + 6x - 112 = 0$$

Factoring the Equation

Now, we can factor the left-hand side of the equation using the factoring method:

$$(x + 14)(x - 8) = 0$$

Solving for x

To find the values of $x$ that satisfy the equation, we need to set each factor equal to zero and solve for $x$. This gives us:

$$x + 14 = 0 \Rightarrow x = -14$$

$$x - 8 = 0 \Rightarrow x = 8$$

Conclusion

In conclusion, the values of $x$ that satisfy the given equation are $x = -14$ and $x = 8$. Therefore, the correct answer is:

B. $x = -14, x = 8$

Discussion

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we focused on solving a specific quadratic equation, $2(x+3)^2 - 242 = 0$, and explored the different values of $x$ that satisfy the given equation. We used various methods, including factoring and completing the square, to solve the equation and find the values of $x$.

Common Mistakes

When solving quadratic equations, there are several common mistakes that students and professionals can make. These include:

• Not following the order of operations: When solving quadratic equations, it's essential to follow the order of operations (PEMDAS) to ensure that the equation is solved correctly.
• Not factoring correctly: Factoring is a crucial step in solving quadratic equations. However, it's easy to make mistakes when factoring, especially when dealing with complex equations.
• Not checking the solutions: When solving quadratic equations, it's essential to check the solutions to ensure that they are correct.

Real-World Applications

Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to optimize economic outcomes.

Conclusion

$$$$$$

Frequently Asked Questions

Q: What is a quadratic equation?

A: A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ is the variable.

Q: How do I solve a quadratic equation?

A: There are several methods to solve quadratic equations, including factoring, completing the square, and the quadratic formula. The method you choose will depend on the specific equation and your personal preference.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that can be used to solve quadratic equations. The formula is:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Q: What is the difference between factoring and completing the square?

A: Factoring and completing the square are two different methods for solving quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves rewriting the quadratic equation in a form that allows for easy solution.

Q: How do I know which method to use?

A: The method you choose will depend on the specific equation and your personal preference. If the equation can be easily factored, then factoring may be the best choice. If the equation cannot be easily factored, then completing the square or the quadratic formula may be a better option.

Q: What are some common mistakes to avoid when solving quadratic equations?

A: Some common mistakes to avoid when solving quadratic equations include:

• Not following the order of operations
• Not factoring correctly
• Not checking the solutions
• Not using the correct method for the specific equation

Q: How do I check my solutions?

A: To check your solutions, plug the values back into the original equation and simplify. If the equation is true, then the solution is correct. If the equation is not true, then the solution is incorrect.

Q: What are some real-world applications of quadratic equations?

A: Quadratic equations have numerous real-world applications, including:

• Physics: Quadratic equations are used to model the motion of objects under the influence of gravity, friction, and other forces.
• Engineering: Quadratic equations are used to design and optimize systems, such as bridges, buildings, and electronic circuits.
• Economics: Quadratic equations are used to model economic systems, including supply and demand curves, and to optimize economic outcomes.

Q: How do I use quadratic equations in real-world problems?

A: To use quadratic equations in real-world problems, identify the variables and constants in the equation, and then use the equation to model the problem. For example, if you are designing a bridge, you can use a quadratic equation to model the stress on the bridge and determine the optimal design.

Q: What are some tips for solving quadratic equations?

A: Some tips for solving quadratic equations include:

• Read the problem carefully and identify the variables and constants.
• Choose the correct method for the specific equation.
• Check your solutions carefully.
• Use a calculator or computer program to check your solutions.

Conclusion

In conclusion, quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. By understanding the different methods for solving quadratic equations, including factoring, completing the square, and the quadratic formula, you can solve quadratic equations with confidence and apply the concepts to real-world problems.