What Number Must You Add To Complete The Square In The Equation $x^2 + 8x = 11$?A. 4 B. 8 C. 16 D. 12

What is Completing the Square?

Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, making it easier to solve. This method involves manipulating the equation to create a perfect square trinomial, which can be factored into a binomial squared. The process of completing the square is essential in algebra, as it allows us to find the solutions to quadratic equations.

The Equation: $x^2 + 8x = 11$

We are given the quadratic equation $x^2 + 8x = 11$. Our goal is to complete the square and find the number that must be added to both sides of the equation to make it a perfect square trinomial.

Step 1: Move the Constant Term to the Right Side

To begin, we need to move the constant term, 11, to the right side of the equation. This will give us:

$$x^2 + 8x - 11 = 0$$

Step 2: Find the Number to Add

To complete the square, we need to find the number that must be added to the left side of the equation. This number is equal to the square of half the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is 8, so we need to find the square of half of 8.

Half of 8 is 4, and the square of 4 is 16. Therefore, the number that must be added to the left side of the equation is 16.

Step 3: Add the Number to Both Sides

Now that we have found the number to add, we can add it to both sides of the equation:

$$x^2 + 8x + 16 = 11 + 16$$

This simplifies to:

$$x^2 + 8x + 16 = 27$$

Step 4: Factor the Left Side

The left side of the equation is now a perfect square trinomial. We can factor it into a binomial squared:

$$(x + 4)^2 = 27$$

Step 5: Solve for $x$

To solve for $x$, we can take the square root of both sides of the equation:

$$x + 4 = \pm \sqrt{27}$$

Simplifying, we get:

$$x + 4 = \pm 3\sqrt{3}$$

Subtracting 4 from both sides, we get:

$$x = -4 \pm 3\sqrt{3}$$

Conclusion

In this article, we have completed the square for the equation $x^2 + 8x = 11$. We found that the number that must be added to both sides of the equation is 16. By completing the square, we were able to rewrite the equation in a form that makes it easier to solve. We hope that this article has provided a clear and concise explanation of the process of completing the square.

Answer

The correct answer is C. 16.

Additional Resources

For more information on completing the square, we recommend checking out the following resources:

• Khan Academy: Completing the Square
• Mathway: Completing the Square
• Wolfram Alpha: Completing the Square

Frequently Asked Questions

Q: What is completing the square? A: Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, making it easier to solve.

Q: How do I complete the square? A: To complete the square, you need to move the constant term to the right side of the equation, find the number to add, add the number to both sides, factor the left side, and solve for $x$.

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Frequently Asked Questions

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite a quadratic equation in a specific form, making it easier to solve. This method involves manipulating the equation to create a perfect square trinomial, which can be factored into a binomial squared.

Q: How do I complete the square?

A: To complete the square, you need to follow these steps:

1. Move the constant term to the right side of the equation.
2. Find the number to add, which is equal to the square of half the coefficient of the $x$ term.
3. Add the number to both sides of the equation.
4. Factor the left side of the equation.
5. Solve for $x$.

Q: What is the number that must be added to complete the square in the equation $x^2 + 8x = 11$?

A: The number that must be added to complete the square in the equation $x^2 + 8x = 11$ is 16.

Q: How do I find the number to add?

A: To find the number to add, you need to take half of the coefficient of the $x$ term and square it. For example, if the coefficient of the $x$ term is 8, you would take half of 8, which is 4, and square it, which gives you 16.

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

A: Completing the square and factoring 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 specific form that makes it easier to solve.

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

A: No, completing the square is not suitable for all types of quadratic equations. This method is only useful for quadratic equations that can be rewritten in the form $x^2 + bx + c = 0$, where $b$ and $c$ are constants.

Q: Are there any other methods for solving quadratic equations?

A: Yes, there are several other methods for solving quadratic equations, including:

• Factoring
• Quadratic formula
• Graphing
• Synthetic division

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 side of the equation
• Not finding the correct number to add
• Not adding the number to both sides of the equation
• Not factoring the left side of the equation correctly

Q: Can I use a calculator to complete the square?

A: Yes, you can use a calculator to complete the square. However, it's often more helpful to do the calculations by hand, as this can help you understand the process better.

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

A: To check if you've completed the square correctly, you can plug the solution back into the original equation and verify that it's true. You can also use a calculator to check your solution.

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

A: Completing the square has several real-world applications, including:

• Physics: Completing the square is used to solve problems involving motion and energy.
• Engineering: Completing the square is used to design and optimize systems.
• Economics: Completing the square is used to model and analyze economic systems.

Conclusion

In this article, we've answered some of the most frequently asked questions about completing the square. We hope that this guide has been helpful in understanding the process of completing the square and how to apply it in different situations.