Solve The Equation For All Values Of { X $}$ By Completing The Square. Express Your Answer In Simplest Form.${ X^2 + 8x = -7 }$

Introduction

Completing the square is a powerful technique used to solve quadratic equations. It 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 how to solve the quadratic equation ${ x^2 + 8x = -7 }$ by completing the square.

Understanding the Quadratic Equation

A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is ${ ax^2 + bx + c = 0 }$, where ${ a }$, ${ b }$, and ${ c }$ are constants. In our case, the equation is ${ x^2 + 8x = -7 }$.

Step 1: Move the Constant Term to the Right Side

To complete the square, we need to move the constant term to the right side of the equation. This gives us:

${ x^2 + 8x + 7 = 0 }$

Step 2: Add and Subtract the Square of Half the Coefficient of x

The coefficient of ${ x }$ is 8. To complete the square, we need to add and subtract the square of half the coefficient of ${ x }$, which is ${ \left( \frac{8}{2} \right)^2 = 16 }$. This gives us:

${ x^2 + 8x + 16 - 16 + 7 = 0 }$

Step 3: Factor the Perfect Square Trinomial

The expression ${ x^2 + 8x + 16 }$ is a perfect square trinomial, which can be factored as:

${ (x + 4)^2 - 16 + 7 = 0 }$

Step 4: Simplify the Equation

Simplifying the equation, we get:

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

Step 5: Add 9 to Both Sides

Adding 9 to both sides of the equation, we get:

${ (x + 4)^2 = 9 }$

Step 6: Take the Square Root of Both Sides

Taking the square root of both sides of the equation, we get:

${ x + 4 = \pm \sqrt{9} }$

Step 7: Simplify the Square Root

Simplifying the square root, we get:

${ x + 4 = \pm 3 }$

Step 8: Solve for x

Solving for ${ x }$, we get:

${ x = -4 \pm 3 }$

Step 9: Simplify the Expression

Simplifying the expression, we get:

${ x = -4 + 3 \quad \text{or} \quad x = -4 - 3 }$

Step 10: Final Answer

The final answer is:

${ x = -1 \quad \text{or} \quad x = -7 }$

Conclusion

In this article, we have solved the quadratic equation ${ x^2 + 8x = -7 }$ by completing the square. We have followed the steps to move the constant term to the right side, add and subtract the square of half the coefficient of ${ x }$, factor the perfect square trinomial, simplify the equation, add 9 to both sides, take the square root of both sides, simplify the square root, solve for ${ x }$, and simplify the expression. The final answer is ${ x = -1 }$ or ${ x = -7 }$.

Example Use Cases

Completing the square is a powerful technique used to solve quadratic equations. It has many applications in mathematics, science, and engineering. Here are a few example use cases:

• Physics: Completing the square is used to solve problems involving motion, such as finding the position and velocity of an object.
• Engineering: Completing the square is used to design and optimize systems, such as electrical circuits and mechanical systems.
• Computer Science: Completing the square is used in algorithms and data structures, such as finding the shortest path in a graph.

Tips and Tricks

Here are a few tips and tricks to help you complete the square:

• Make sure to move the constant term to the right side: This is the first step in completing the square.
• Add and subtract the square of half the coefficient of x: This is the key step in completing the square.
• Factor the perfect square trinomial: This will help you simplify the equation.
• Simplify the equation: This will help you find the final answer.

Conclusion

Q: What is completing the square?

A: Completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in a perfect square trinomial form.

Q: Why is completing the square useful?

A: Completing the square is useful because it allows us to solve quadratic equations in a straightforward and efficient manner. It is also a powerful technique used in many areas of mathematics, science, and engineering.

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 side of the equation.
2. Add and subtract the square of half the coefficient of x.
3. Factor the perfect square trinomial.
4. Simplify the equation.
5. Add 9 to both sides.
6. Take the square root of both sides.
7. Simplify the square root.
8. Solve for x.

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 manipulating the equation to express it in a perfect square trinomial 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 solutions to the equation in a specific form.

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 adding and subtracting the square of half the coefficient of x.
• Not factoring the perfect square trinomial.
• Not simplifying the equation.
• Not adding 9 to both sides.
• Not taking the square root of both sides.
• Not simplifying the square root.

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 and checking if they are true.

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 will need to use complex numbers and complex arithmetic.

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

A: Yes, you can use completing the square to solve quadratic equations with rational coefficients. However, you will need to use rational numbers and rational arithmetic.

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

A: Yes, you can use completing the square to solve quadratic equations with irrational coefficients. However, you will need to use irrational numbers and irrational arithmetic.

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

A: Yes, you can use completing the square to solve quadratic equations with mixed coefficients. However, you will need to use a combination of rational and irrational numbers and arithmetic.

Conclusion

In conclusion, completing the square is a powerful technique used to solve quadratic equations. It involves manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. By following the steps outlined in this article, you can solve quadratic equations by completing the square.