Which Of The Following Correctly Completes The Square For The Equation Below?$x^2 + 4x = 14$A. $(x+2)^2 = 30$ B. $(x+2)^2 = 18$ C. $(x+4)^2 = 30$ D. $(x+4)^2 = 18$

Introduction

Completing the square is a powerful algebraic technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square form, which can be easily solved. In this article, we will explore the concept of completing the square and apply it to the given equation $x^2 + 4x = 14$.

What is Completing the Square?

Completing the square is a method of solving quadratic equations by rewriting them in a perfect square form. This involves adding and subtracting a constant term to create a perfect square trinomial. The general form of a quadratic equation is $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants.

The Process of Completing the Square

To complete the square, we follow these steps:

1. Move the constant term to the right-hand side: We move the constant term $c$ to the right-hand side of the equation, so that the equation becomes $ax^2 + bx = -c$.
2. Add and subtract $(b/2)^2$: We add and subtract $(b/2)^2$ to the left-hand side of the equation. This creates a perfect square trinomial.
3. Factor the perfect square trinomial: We factor the perfect square trinomial to obtain the square of a binomial.
4. Solve for x: We solve for $x$ by taking the square root of both sides of the equation.

Applying Completing the Square to the Given Equation

Now, let's apply the process of completing the square to the given equation $x^2 + 4x = 14$.

Step 1: Move the constant term to the right-hand side

We move the constant term $14$ to the right-hand side of the equation, so that the equation becomes $x^2 + 4x - 14 = 0$.

Step 2: Add and subtract $(b/2)^2$

We add and subtract $(4/2)^2 = 4$ to the left-hand side of the equation. This creates a perfect square trinomial.

$x^2 + 4x + 4 - 4 - 14 = 0$

Step 3: Factor the perfect square trinomial

We factor the perfect square trinomial to obtain the square of a binomial.

$(x + 2)^2 - 18 = 0$

Step 4: Solve for x

We solve for $x$ by taking the square root of both sides of the equation.

$(x + 2)^2 = 18$

Which of the Following Correctly Completes the Square?

Now, let's compare our result with the given options.

• A. $(x+2)^2 = 30$
• B. $(x+2)^2 = 18$
• C. $(x+4)^2 = 30$
• D. $(x+4)^2 = 18$

Our result matches option B. Therefore, the correct answer is:

B. $(x+2)^2 = 18$

Conclusion

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Introduction

Completing the square is a powerful algebraic technique used to solve quadratic equations. In our previous article, we explored the concept of completing the square and applied it to the given equation $x^2 + 4x = 14$. In this article, we will answer some frequently asked questions about completing the square.

Q&A

Q: What is the purpose of completing the square?

A: The purpose of completing the square is to rewrite a quadratic equation in a perfect square form, which can be easily solved.

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

A: The steps involved in completing the square are:

1. Move the constant term to the right-hand side: We move the constant term $c$ to the right-hand side of the equation, so that the equation becomes $ax^2 + bx = -c$.
2. Add and subtract $(b/2)^2$: We add and subtract $(b/2)^2$ to the left-hand side of the equation. This creates a perfect square trinomial.
3. Factor the perfect square trinomial: We factor the perfect square trinomial to obtain the square of a binomial.
4. Solve for x: We solve for $x$ by taking the square root of both sides of the equation.

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

A: Factoring involves expressing a quadratic equation as a product of two binomials, while completing the square involves rewriting a quadratic equation in a perfect square form.

Q: When should I use completing the square?

A: You should use completing the square when the quadratic equation does not factor easily, or when you need to find the vertex of a parabola.

Q: Can I use completing the square to solve quadratic equations with complex coefficients?

A: Yes, you can use completing the square to solve quadratic equations with complex coefficients.

Q: How do I know if I have completed the square correctly?

A: You can check if you have completed the square correctly by plugging the solution back into the original equation.

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

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

• Not moving the constant term to the right-hand side
• Not adding and subtracting $(b/2)^2$ correctly
• Not factoring the perfect square trinomial correctly
• Not solving for $x$ correctly

Examples and Practice Problems

Example 1: Completing the Square for $x^2 + 6x = 12$

We can complete the square for the equation $x^2 + 6x = 12$ by following the steps outlined above.

Step 1: Move the constant term to the right-hand side

We move the constant term $12$ to the right-hand side of the equation, so that the equation becomes $x^2 + 6x - 12 = 0$.

Step 2: Add and subtract $(b/2)^2$

We add and subtract $(6/2)^2 = 9$ to the left-hand side of the equation. This creates a perfect square trinomial.

$x^2 + 6x + 9 - 9 - 12 = 0$

Step 3: Factor the perfect square trinomial

We factor the perfect square trinomial to obtain the square of a binomial.

$(x + 3)^2 - 21 = 0$

Step 4: Solve for x

We solve for $x$ by taking the square root of both sides of the equation.

$(x + 3)^2 = 21$

Example 2: Completing the Square for $x^2 - 4x = 8$

We can complete the square for the equation $x^2 - 4x = 8$ by following the steps outlined above.

Step 1: Move the constant term to the right-hand side

We move the constant term $8$ to the right-hand side of the equation, so that the equation becomes $x^2 - 4x - 8 = 0$.

Step 2: Add and subtract $(b/2)^2$

We add and subtract $(-4/2)^2 = 4$ to the left-hand side of the equation. This creates a perfect square trinomial.

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

Step 3: Factor the perfect square trinomial

We factor the perfect square trinomial to obtain the square of a binomial.

$(x - 2)^2 - 12 = 0$

Step 4: Solve for x

We solve for $x$ by taking the square root of both sides of the equation.

$(x - 2)^2 = 12$

Conclusion

In this article, we answered some frequently asked questions about completing the square. We also provided examples and practice problems to help you understand the concept better. Completing the square is a powerful algebraic technique used to solve quadratic equations, and it has many applications in mathematics and science.