Complete The Square To Solve $3x^2 + 12x = 9$.A. $x = -2 \pm \sqrt{13}$B. \$x = 2 \pm \sqrt{7}$[/tex\]C. $x = -2 \pm \sqrt{7}$D. $x = 3 \pm \sqrt{6}$

Introduction

Solving quadratic equations is a fundamental concept in mathematics, and one of the most effective methods for solving these equations is by completing the square. This method involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we will explore the concept of completing the square and apply it to solve the quadratic equation $3x^2 + 12x = 9$.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The method involves manipulating the equation to express it in the form $(x + p)^2 = q$, where $p$ and $q$ are constants. This form can be easily solved by taking the square root of both sides of the equation.

Step 1: Move the Constant Term to the Right-Hand Side

The first step in completing the square is to move the constant term to the right-hand side of the equation. In this case, we have $3x^2 + 12x = 9$. To move the constant term to the right-hand side, we subtract 9 from both sides of the equation:

$$3x^2 + 12x - 9 = 0$$

Step 2: Factor Out the Coefficient of the $x^2$ Term

The next step is to factor out the coefficient of the $x^2$ term. In this case, the coefficient of the $x^2$ term is 3. We factor out 3 from the first two terms of the equation:

$$3(x^2 + 4x) - 9 = 0$$

Step 3: Add and Subtract the Square of Half the Coefficient of the $x$ Term

The next step is to add and subtract the square of half the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is 4. Half of 4 is 2, and the square of 2 is 4. We add and subtract 4 inside the parentheses:

$$3(x^2 + 4x + 4 - 4) - 9 = 0$$

Step 4: Simplify the Equation

We simplify the equation by combining like terms:

$$3(x^2 + 4x + 4) - 12 - 9 = 0$$

$$3(x^2 + 4x + 4) = 21$$

Step 5: Factor the Perfect Square Trinomial

The next step is to factor the perfect square trinomial:

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

Step 6: Divide Both Sides by 3

We divide both sides of the equation by 3 to isolate the perfect square trinomial:

$$(x + 2)^2 = 7$$

Step 7: Take the Square Root of Both Sides

We take the square root of both sides of the equation to solve for $x$:

$$x + 2 = \pm \sqrt{7}$$

Step 8: Solve for $x$

We solve for $x$ by subtracting 2 from both sides of the equation:

$$x = -2 \pm \sqrt{7}$$

Conclusion

In this article, we have explored the concept of completing the square and applied it to solve the quadratic equation $3x^2 + 12x = 9$. We have followed the steps of moving the constant term to the right-hand side, factoring out the coefficient of the $x^2$ term, adding and subtracting the square of half the coefficient of the $x$ term, simplifying the equation, factoring the perfect square trinomial, dividing both sides by 3, taking the square root of both sides, and solving for $x$. The solution to the equation is $x = -2 \pm \sqrt{7}$.

Answer

The correct answer is:

• A. $x = -2 \pm \sqrt{7}$
  Completing the Square: A Q&A Guide =====================================

Introduction

Completing the square is a powerful technique used to solve quadratic equations. In our previous article, we explored the concept of completing the square and applied it to solve the quadratic equation $3x^2 + 12x = 9$. In this article, we will answer some frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations of the form $ax^2 + bx + c = 0$. The method involves manipulating the equation to express it in the form $(x + p)^2 = q$, where $p$ and $q$ are constants.

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

A: You should use completing the square when the quadratic equation is not easily factorable, or when you need to find the solutions to the equation.

Q: What are the steps to complete the square?

A: The steps to complete the square are:

1. Move the constant term to the right-hand side of the equation.
2. Factor out the coefficient of the $x^2$ term.
3. Add and subtract the square of half the coefficient of the $x$ term.
4. Simplify the equation.
5. Factor the perfect square trinomial.
6. Divide both sides by the coefficient of the perfect square trinomial.
7. Take the square root of both sides of the equation.
8. Solve for $x$.

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 expressing a quadratic equation in the form $(x + p)^2 = q$.

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

A: Yes, you can use completing the square to solve all types of quadratic equations, including those that are not easily factorable.

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 of the equation.
• Not factoring out the coefficient of the $x^2$ term.
• Not adding and subtracting the square of half the coefficient of the $x$ term.
• Not simplifying the equation.
• Not factoring the perfect square trinomial.
• Not dividing both sides by the coefficient of the perfect square trinomial.
• Not taking the square root of both sides of the equation.

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 solutions back into the original equation. If the solutions satisfy the equation, then you have completed the square correctly.

Conclusion

In this article, we have answered some frequently asked questions about completing the square. We have discussed the concept of completing the square, the steps to complete the square, and some common mistakes to avoid. We have also provided some tips on how to check if you have completed the square correctly.

Additional Resources

If you are interested in learning more about completing the square, we recommend checking out the following resources:

• Khan Academy: Completing the Square
• Mathway: Completing the Square
• Purplemath: Completing the Square

Practice Problems

Here are some practice problems to help you practice completing the square:

• Solve the quadratic equation $x^2 + 6x + 8 = 0$ using completing the square.
• Solve the quadratic equation $2x^2 + 4x + 3 = 0$ using completing the square.
• Solve the quadratic equation $x^2 - 4x + 4 = 0$ using completing the square.

We hope this article has been helpful in answering your questions about completing the square. If you have any further questions, please don't hesitate to ask.