Solve The Equation For All Values Of \[$ X \$\] By Completing The Square. Express Your Answer In Simplest Form.$\[ X^2 + 20x = -97 \\]

Feb 27, 2025 by ADMIN 135 views

**Solving Quadratic Equations by Completing the Square**

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 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 how to solve the equation ${ x^2 + 20x = -97 }$ by completing the square.

Understanding the Concept of Completing the Square

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

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:

${ x^2 + 20x = -97 }$

To move the constant term to the right-hand side, we add 97 to both sides of the equation:

${ x^2 + 20x + 97 = 0 }$

Step 2: Find the Value to Add to Both Sides

The next step is to find the value to add to both sides of the equation to make the left-hand side a perfect square trinomial. To do this, we need to find the value of ${ p }$ such that ${ (x + p)^2 = x^2 + 20x + p^2 }$ . We can see that ${ p^2 = 100 }$ , so ${ p = 10 }$ or ${ p = -10 }$ . Since we are adding a positive value to both sides, we choose ${ p = 10 }$ .

Step 3: Add the Value to Both Sides

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

${ x^2 + 20x + 100 = 97 }$

Step 4: Factor the Left-Hand Side

The left-hand side of the equation is now a perfect square trinomial. We can factor it as follows:

${ (x + 10)^2 = 97 }$

Step 5: Take the Square Root of Both Sides

Finally, we can take the square root of both sides of the equation to solve for ${ x }$ :

${ x + 10 = \pm \sqrt{97} }$

Simplifying the Solution

We can simplify the solution by subtracting 10 from both sides of the equation:

${ x = -10 \pm \sqrt{97} }$

Conclusion

In this article, we have shown how to solve the equation ${ x^2 + 20x = -97 }$ by completing the square. We have broken down the solution into five steps, and we have provided a detailed explanation of each step. By following these steps, we can solve quadratic equations of the form ${ ax^2 + bx + c = 0 }$ by completing the square.

Tips and Variations

To solve quadratic equations of the form ${ ax^2 + bx + c = 0 }$ , we can use the completing the square method.
The completing the square method involves manipulating the equation to express it in the form ${ a(x + p)^2 + q = 0 }$ , where ${ p }$ and ${ q }$ are constants.
To find the value to add to both sides of the equation, we need to find the value of ${ p }$ such that ${ (x + p)^2 = x^2 + 20x + p^2 }$ .
We can add the value to both sides of the equation to make the left-hand side a perfect square trinomial.
We can factor the left-hand side of the equation as follows: ${ (x + p)^2 = 97 }$ .
We can take the square root of both sides of the equation to solve for ${ x }$ : ${ x + 10 = \pm \sqrt{97} }$ .
We can simplify the solution by subtracting 10 from both sides of the equation: ${ x = -10 \pm \sqrt{97} }$ .

Common Mistakes to Avoid

When solving quadratic equations by completing the square, it is easy to make mistakes. One common mistake is to forget to add the value to both sides of the equation.
Another common mistake is to forget to factor the left-hand side of the equation.
To avoid these mistakes, it is essential to follow the steps carefully and to check the solution at each step.

Real-World Applications

Quadratic equations have many real-world applications. For example, they can be used to model the motion of objects under the influence of gravity.
They can also be used to model the growth of populations and the spread of diseases.
In addition, quadratic equations can be used to solve problems in physics, engineering, and economics.

Conclusion

Introduction

In our previous article, we explored how to solve quadratic equations 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 answer some of the most frequently asked questions about solving quadratic equations by 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 idea behind this method is to manipulate the equation to express it in the form ${ a(x + p)^2 + q = 0 }$ , 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 in the form ${ ax^2 + bx + c = 0 }$ and you want to solve for ${ x }$ . This method is particularly useful when the equation is not easily factorable.

Q: What are the steps to complete the square?

A: The steps to complete the square are as follows:

Move the constant term to the right-hand side of the equation.
Find the value to add to both sides of the equation to make the left-hand side a perfect square trinomial.
Add the value to both sides of the equation.
Factor the left-hand side of the equation.
Take the square root of both sides of the equation to solve for ${ x }$ .

Q: How do I find the value to add to both sides of the equation?

A: To find the value to add to both sides of the equation, you need to find the value of ${ p }$ such that ${ (x + p)^2 = x^2 + 20x + p^2 }$ . You can do this by taking half of the coefficient of the ${ x }$ term and squaring it.

Q: What if I get a negative value when I take the square root of both sides of the equation?

A: If you get a negative value when you take the square root of both sides of the equation, it means that the equation has no real solutions. In this case, you can ignore the negative value and only consider the positive value.

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 taking the square root of both sides of the equation, as this can introduce complex numbers.

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

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

Forgetting to add the value to both sides of the equation.
Forgetting to factor the left-hand side of the equation.
Taking the square root of both sides of the equation without checking if the equation has real solutions.

Q: How can I practice completing the square?

A: You can practice completing the square by working through examples and exercises. You can also use online resources and practice tests to help you improve your skills.

Conclusion

In conclusion, solving quadratic equations by completing the square is a powerful technique that can be used to solve a wide range of problems. By following the steps carefully and checking the solution at each step, we can avoid common mistakes and arrive at the correct solution. Whether we are solving quadratic equations in mathematics or applying them to real-world problems, completing the square is an essential tool that we need to master.

Additional Resources

Frequently Asked Questions

Q: What is completing the square? A: Completing the square is a technique used to solve quadratic equations of the form ${ ax^2 + bx + c = 0 }$ .
Q: How do I know when to use completing the square? A: You should use completing the square when the quadratic equation is in the form ${ ax^2 + bx + c = 0 }$ and you want to solve for ${ x }$ .
Q: What are the steps to complete the square? A: The steps to complete the square are as follows:
1. Move the constant term to the right-hand side of the equation.
2. Find the value to add to both sides of the equation to make the left-hand side a perfect square trinomial.
3. Add the value to both sides of the equation.
4. Factor the left-hand side of the equation.
5. Take the square root of both sides of the equation to solve for ${ x }$ .