Not adding and subtracting the constant of completion correctly

Oct 13, 2025 by miodeya.com 65 views

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 manipulating the equation to express it in a perfect square trinomial form, which can then be easily solved. In this article, we will focus on solving the quadratic equation $x^2 + 4x - 21 = 0$ by completing the square.

What is Completing the Square?

Completing the square is a technique used to solve quadratic equations by manipulating the equation to express it in a perfect square trinomial form. This involves adding and subtracting a constant term to the equation, which allows us to express the quadratic expression as a perfect square. The constant term added is called the "constant of completion," and it is equal to the square of half the coefficient of the linear term.

Step 1: Write the Quadratic Equation

The given quadratic equation is $x^2 + 4x - 21 = 0$ . To solve this equation by completing the square, we need to write it in the standard form of a quadratic equation, which is $ax^2 + bx + c = 0$ .

Step 2: Identify the Coefficients

In the given equation, the coefficient of the quadratic term is $a = 1$ , the coefficient of the linear term is $b = 4$ , and the constant term is $c = -21$ .

Step 3: Calculate the Constant of Completion

The constant of completion is equal to the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is $b = 4$ , so half of it is $2$ . The square of $2$ is $4$ , so the constant of completion is $4$ .

Step 4: Add and Subtract the Constant of Completion

To complete the square, we need to add and subtract the constant of completion to the equation. This gives us:

$x^2 + 4x + 4 - 4 - 21 = 0$

Step 5: Express the Quadratic Expression as a Perfect Square

The first three terms of the equation form a perfect square trinomial:

$(x + 2)^2 - 25 = 0$

Step 6: Solve for x

Now that we have expressed the quadratic expression as a perfect square, we can easily solve for $x$ . We can start by adding $25$ to both sides of the equation:

$(x + 2)^2 = 25$

Step 7: Take the Square Root

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

$x + 2 = \pm \sqrt{25}$

Step 8: Simplify

The square root of $25$ is $5$ , so we can simplify the equation:

$x + 2 = \pm 5$

Step 9: Solve for x

To solve for $x$ , we need to isolate the variable. We can do this by subtracting $2$ from both sides of the equation:

$x = -2 \pm 5$

Step 10: Simplify

The solutions to the equation are:

$x = -2 + 5 = 3$

$x = -2 - 5 = -7$

Conclusion

In this article, we have solved the quadratic equation $x^2 + 4x - 21 = 0$ by completing the square. We have shown that the solutions to the equation are $x = 3$ and $x = -7$ . This method of solving quadratic equations is a powerful tool that can be used to solve a wide range of equations.

Why is Completing the Square Important?

Completing the square is an important technique in mathematics because it allows us to solve quadratic equations in a way that is easy to understand and visualize. It is also a useful technique for solving equations that are not easily solvable using other methods. Additionally, completing the square is a fundamental concept in algebra and is used extensively in calculus and other areas of mathematics.

Real-World Applications of Completing the Square

Completing the square has many real-world applications. For example, it is used in physics to solve equations of motion, in engineering to design and optimize systems, and in economics to model and analyze economic systems. It is also used in computer science to solve problems in algorithms and data structures.

Common Mistakes to Avoid

When completing the square, there are several common mistakes to avoid. These include:

Not adding and subtracting the constant of completion correctly: Make sure to add and subtract the constant of completion correctly to avoid errors.
Not expressing the quadratic expression as a perfect square: Make sure to express the quadratic expression as a perfect square to avoid errors.
Not solving for x correctly: Make sure to solve for x correctly by isolating the variable.

Conclusion

Introduction

In our previous article, we discussed 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 common questions related to completing the square.

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. This involves adding and subtracting a constant term to the equation, which allows us to express the quadratic expression as a perfect square.

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 solvable using other methods, such as factoring or the quadratic formula. Completing the square is particularly useful when the equation has a coefficient of 1 on the quadratic term.

Q: What is the constant of completion?

A: The constant of completion is the term that is added and subtracted to the equation to complete the square. It is equal to the square of half the coefficient of the linear term.

Q: How do I calculate the constant of completion?

A: To calculate the constant of completion, you need to find half the coefficient of the linear term and then square it. For example, if the coefficient of the linear term is 4, half of it is 2, and the square of 2 is 4.

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 expression as a product of two binomials, while completing the square involves expressing the quadratic expression as a perfect square trinomial.

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

A: No, completing the square is not suitable for all quadratic equations. It is particularly useful for equations with a coefficient of 1 on the quadratic term. For equations with a coefficient other than 1, you may need to use other methods, such as the quadratic formula.

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

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

Not adding and subtracting the constant of completion correctly
Not expressing the quadratic expression as a perfect square
Not solving for x correctly by isolating the variable

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

A: Yes, completing the square can be used to solve equations with complex coefficients. However, you need to be careful when working with complex numbers and make sure to follow the correct procedures.

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

A: To check if you have completed the square correctly, you can plug the solution back into the original equation and verify that it is true. You can also use the fact that the solution should be in the form of a perfect square trinomial.

Conclusion

In conclusion, completing the square is a powerful technique for solving quadratic equations. By understanding and mastering this technique, you can solve a wide range of equations and problems in mathematics and other areas of study. We hope that this Q&A article has helped to clarify any questions you may have had about completing the square.

Additional Resources

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

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

Practice Problems

To practice completing the square, try solving the following problems:

$x^2 + 6x - 8 = 0$
$x^2 - 4x - 5 = 0$
$x^2 + 2x - 3 = 0$

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