Consider The Equation:${ X^2 + 10x + 22 = 13 }$1) Rewrite The Equation By Completing The Square.Your Equation Should Look Like { (x+c)^2 = D$}$ Or { (x-c)^2 = D$}$.

Introduction

Quadratic equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. One of the methods used to solve quadratic equations is by completing the square. This method involves rewriting the equation in a specific form, which allows us to easily identify the solutions. In this article, we will focus on rewriting the equation $x^2 + 10x + 22 = 13$ by completing the square.

Understanding the Method

Completing the square is a technique used to rewrite a quadratic equation in the form $(x+c)^2 = d$ or $(x-c)^2 = d$. This form is useful because it allows us to easily identify the solutions to the equation. To complete the square, we need to follow a few steps:

1. Move the constant term to the right-hand side of the equation.
2. Take half of the coefficient of the $x$ term and square it.
3. Add the result to both sides of the equation.
4. Factor the left-hand side of the equation as a perfect square.

Rewriting the Equation

Let's apply the steps above to rewrite the equation $x^2 + 10x + 22 = 13$ by completing the square.

Step 1: Move the Constant Term

The first step is to move the constant term to the right-hand side of the equation. We can do this by subtracting 13 from both sides of the equation.

$$x^2 + 10x + 22 - 13 = 13 - 13$$

This simplifies to:

$$x^2 + 10x + 9 = 0$$

Step 2: Take Half of the Coefficient of the $x$ Term

The next step is to take half of the coefficient of the $x$ term. In this case, the coefficient of the $x$ term is 10, so we take half of that, which is 5.

Step 3: Square the Result

We then square the result, which gives us 25.

Step 4: Add the Result to Both Sides

Now, we add 25 to both sides of the equation.

$$x^2 + 10x + 25 = 9 + 25$$

This simplifies to:

$$(x+5)^2 = 34$$

Step 5: Factor the Left-Hand Side

Finally, we factor the left-hand side of the equation as a perfect square.

$$(x+5)^2 = 34$$

This is the final form of the equation, which is in the form $(x+c)^2 = d$.

Conclusion

In this article, we have learned how to rewrite the equation $x^2 + 10x + 22 = 13$ by completing the square. We have followed the steps to move the constant term, take half of the coefficient of the $x$ term, square the result, add the result to both sides, and factor the left-hand side as a perfect square. The final form of the equation is $(x+5)^2 = 34$, which allows us to easily identify the solutions to the equation.

Solutions to the Equation

To find the solutions to the equation, we can take the square root of both sides of the equation.

$$x+5 = \pm\sqrt{34}$$

This gives us two possible solutions:

$$x = -5 + \sqrt{34}$$

$$x = -5 - \sqrt{34}$$

These are the solutions to the equation $x^2 + 10x + 22 = 13$.

Real-World Applications

Completing the square has many real-world applications. For example, it can be used to solve quadratic equations that arise in physics, engineering, and economics. It can also be used to find the maximum or minimum value of a quadratic function.

Conclusion

$$$$$$$$$$

Introduction

Completing the square is a powerful technique used to solve quadratic equations. In our previous article, we learned how to rewrite the equation $x^2 + 10x + 22 = 13$ by completing the square. In this article, we will answer some of the most frequently asked questions about completing the square.

Q: What is completing the square?

A: Completing the square is a technique used to rewrite a quadratic equation in the form $(x+c)^2 = d$ or $(x-c)^2 = d$. This form is useful because it allows us to easily identify the solutions to the equation.

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-hand side of the equation.
2. Take half of the coefficient of the $x$ term and square it.
3. Add the result to both sides of the equation.
4. Factor the left-hand side of the equation as a perfect square.

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

A: Completing the square and factoring are two different techniques used to solve quadratic equations. Factoring involves expressing the quadratic equation as a product of two binomials, while completing the square involves rewriting the equation in the form $(x+c)^2 = d$ or $(x-c)^2 = d$.

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 solutions to the equation in a specific form.

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. However, you need to be careful when working with complex numbers, as they can be tricky to handle.

Q: How do I find the solutions to the equation after completing the square?

A: To find the solutions to the equation after completing the square, you need to take the square root of both sides of the equation. This will give you two possible solutions.

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

A: Completing the square has many real-world applications. For example, it can be used to solve quadratic equations that arise in physics, engineering, and economics. It can also be used to find the maximum or minimum value of a quadratic function.

Q: Can I use completing the square to solve quadratic inequalities?

A: Yes, you can use completing the square to solve quadratic inequalities. However, you need to be careful when working with inequalities, as they can be tricky to handle.

Q: How do I know if I should use completing the square or factoring to solve a quadratic equation?

A: You should use completing the square if the quadratic equation does not factor easily, or if you need to find the solutions to the equation in a specific form. On the other hand, you should use factoring if the quadratic equation can be easily expressed as a product of two binomials.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. By following the steps outlined in this article, you can answer some of the most frequently asked questions about completing the square. Remember to use completing the square when the quadratic equation does not factor easily, or when you need to find the solutions to the equation in a specific form.

Additional Resources

If you want to learn more about completing the square, here are some additional resources:

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

Practice Problems

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

• Solve the equation $x^2 + 6x + 8 = 0$ by completing the square.
• Solve the equation $x^2 - 4x + 4 = 0$ by completing the square.
• Solve the equation $x^2 + 2x + 1 = 0$ by completing the square.

Answer Key

Here are the answers to the practice problems:

• $x = -3 \pm \sqrt{5}$
• $x = 2$
• $x = -1$