Select The Correct Answer.Solve The Equation Using The Method Of Completing The Square.$2x^2 + 16x - 8 = 0$A. $x = 2 \pm 4 \sqrt{5}$B. $x = -4 \pm 2 \sqrt{5}$C. $x = 4 \pm 2 \sqrt{5}$D. $x = -2 \pm 4 \sqrt{5}$

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. One of the methods used to solve quadratic equations is the method of completing the square. This method involves manipulating the equation to create a perfect square trinomial, which can then be factored to find the solutions. In this article, we will use the method of completing the square to solve the equation $2x^2 + 16x - 8 = 0$.

The Method of Completing the Square

The method of completing the square is a powerful technique for solving quadratic equations. It involves manipulating the equation to create a perfect square trinomial, which can then be factored to find the solutions. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

To complete the square, we need to follow these steps:

1. Divide the equation by the coefficient of the $x^2$ term: If the coefficient of the $x^2$ term is not 1, we need to divide the entire equation by this coefficient to make it 1.
2. Move the constant term to the right-hand side: We need to move the constant term to the right-hand side of the equation to isolate the terms involving $x$.
3. Add and subtract the square of half the coefficient of the $x$ term: We need to add and subtract the square of half the coefficient of the $x$ term to create a perfect square trinomial.
4. Factor the perfect square trinomial: Once we have created a perfect square trinomial, we can factor it to find the solutions.

Solving the Equation $2x^2 + 16x - 8 = 0$

Now that we have understood the method of completing the square, let's apply it to solve the equation $2x^2 + 16x - 8 = 0$.

Step 1: Divide the equation by the coefficient of the $x^2$ term

The coefficient of the $x^2$ term is 2, so we need to divide the entire equation by 2 to make it 1.

$$\frac{2x^2 + 16x - 8}{2} = 0$$

This simplifies to:

$$x^2 + 8x - 4 = 0$$

Step 2: Move the constant term to the right-hand side

We need to move the constant term to the right-hand side of the equation to isolate the terms involving $x$.

$$x^2 + 8x = 4$$

Step 3: Add and subtract the square of half the coefficient of the $x$ term

The coefficient of the $x$ term is 8, so half of this is 4. The square of 4 is 16, so we need to add and subtract 16 to create a perfect square trinomial.

$$x^2 + 8x + 16 = 4 + 16$$

This simplifies to:

$$(x + 4)^2 = 20$$

Step 4: Factor the perfect square trinomial

Now that we have created a perfect square trinomial, we can factor it to find the solutions.

$$(x + 4)^2 = 20$$

Taking the square root of both sides, we get:

$$x + 4 = \pm \sqrt{20}$$

Simplifying the square root, we get:

$$x + 4 = \pm 2\sqrt{5}$$

Subtracting 4 from both sides, we get:

$$x = -4 \pm 2\sqrt{5}$$

Conclusion

In this article, we used the method of completing the square to solve the equation $2x^2 + 16x - 8 = 0$. We divided the equation by the coefficient of the $x^2$ term, moved the constant term to the right-hand side, added and subtracted the square of half the coefficient of the $x$ term, and factored the perfect square trinomial to find the solutions. The solutions to the equation are $x = -4 \pm 2\sqrt{5}$.

Answer

The correct answer is:

• B. $x = -4 \pm 2 \sqrt{5}$

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Q: What is the method of completing the square?

A: The method of completing the square is a technique used to solve quadratic equations by manipulating the equation to create a perfect square trinomial, which can then be factored to find the solutions.

Q: What are the steps involved in completing the square?

A: The steps involved in completing the square are:

1. Divide the equation by the coefficient of the $x^2$ term: If the coefficient of the $x^2$ term is not 1, we need to divide the entire equation by this coefficient to make it 1.
2. Move the constant term to the right-hand side: We need to move the constant term to the right-hand side of the equation to isolate the terms involving $x$.
3. Add and subtract the square of half the coefficient of the $x$ term: We need to add and subtract the square of half the coefficient of the $x$ term to create a perfect square trinomial.
4. Factor the perfect square trinomial: Once we have created a perfect square trinomial, we can factor it to find the solutions.

Q: How do I know when to use the method of completing the square?

A: You should use the method of completing the square when you are given a quadratic equation in the form $ax^2 + bx + c = 0$ and you want to find the solutions.

Q: What are some common mistakes to avoid when completing the square?

A: Some common mistakes to avoid when completing the square include:

• Not dividing the equation by the coefficient of the $x^2$ term: If the coefficient of the $x^2$ term is not 1, you need to divide the entire equation by this coefficient to make it 1.
• Not moving the constant term to the right-hand side: You need to move the constant term to the right-hand side of the equation to isolate the terms involving $x$.
• Not adding and subtracting the square of half the coefficient of the $x$ term: You need to add and subtract the square of half the coefficient of the $x$ term to create a perfect square trinomial.
• Not factoring the perfect square trinomial: Once you have created a perfect square trinomial, you need to factor it to find the solutions.

Q: Can I use the method of completing the square to solve all types of quadratic equations?

A: No, you cannot use the method of completing the square to solve all types of quadratic equations. The method of completing the square is only used to solve quadratic equations that can be written in the form $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

Q: What are some real-world applications of the method of completing the square?

A: The method of completing the square has many real-world applications, including:

• Physics: The method of completing the square is used to solve problems involving motion, energy, and momentum.
• Engineering: The method of completing the square is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: The method of completing the square is used in algorithms and data structures, such as sorting and searching.

Q: How can I practice using the method of completing the square?

A: You can practice using the method of completing the square by working through examples and exercises in a textbook or online resource. You can also try solving quadratic equations on your own using the method of completing the square.

Conclusion

In this article, we have answered some frequently asked questions about solving quadratic equations using the method of completing the square. We have covered topics such as the steps involved in completing the square, common mistakes to avoid, and real-world applications of the method. We hope that this article has been helpful in understanding the method of completing the square and how to use it to solve quadratic equations.