Importance of Completing the Square

Oct 13, 2025 by windailys.com 129 views

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Introduction

Completing the square is a powerful algebraic 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 focus on solving the quadratic equation $r^2 + 24r = -1$ using the completing the square method.

What is Completing the Square?

Completing the square is a method of solving quadratic equations of the form $ax^2 + bx + c = 0$ . The method involves manipulating the equation to express it in the form $(x + \frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b}{2a})^2$ . 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 Side

The first step in completing the square is to move the constant term to the right side of the equation. In this case, we have:

$r^2 + 24r = -1$

We can rewrite this equation as:

$r^2 + 24r + 1 = 0$

Step 2: Add and Subtract the Square of Half the Coefficient of the Linear Term

The next step is to add and subtract the square of half the coefficient of the linear term. In this case, the coefficient of the linear term is 24, so we need to add and subtract $(24/2)^2 = 144$ .

$r^2 + 24r + 144 - 144 + 1 = 0$

Step 3: Factor the Perfect Square Trinomial

Now we can factor the perfect square trinomial:

$(r + 12)^2 - 145 = 0$

Step 4: Add 145 to Both Sides

Next, we add 145 to both sides of the equation:

$(r + 12)^2 = 145$

Step 5: Take the Square Root of Both Sides

Finally, we take the square root of both sides of the equation:

$r + 12 = \pm \sqrt{145}$

Step 6: Solve for r

Now we can solve for r by subtracting 12 from both sides:

$r = -12 \pm \sqrt{145}$

Conclusion

In this article, we have solved the quadratic equation $r^2 + 24r = -1$ using the completing the square method. We have shown that the solution is $r = -12 \pm \sqrt{145}$ . This solution can be expressed as a decimal rounded to the nearest hundredth.

Final Answer

The final answer is: $\boxed{-12 \pm 12.04}$

Discussion

Completing the square is a powerful algebraic 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 have shown how to solve the quadratic equation $r^2 + 24r = -1$ using the completing the square method. We have also discussed the importance of completing the square in algebra and provided a step-by-step guide on how to complete the square.

Importance of Completing the Square

Completing the square is an important algebraic 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. Completing the square is used in a variety of applications, including physics, engineering, and computer science.

Applications of Completing the Square

Completing the square has a wide range of applications in various fields. Some of the applications of completing the square include:

Physics: Completing the square is used to solve problems involving motion, energy, and momentum.
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 sorting and searching.

Conclusion

In conclusion, completing the square is a powerful algebraic 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 have shown how to solve the quadratic equation $r^2 + 24r = -1$ using the completing the square method. We have also discussed the importance of completing the square in algebra and provided a step-by-step guide on how to complete the square.

Final Thoughts

Completing the square is a fundamental concept in algebra that has a wide range of applications in various fields. It is an important technique used to solve quadratic equations and is used in a variety of applications, including physics, engineering, and computer science. In this article, we have shown how to solve the quadratic equation $r^2 + 24r = -1$ using the completing the square method and have discussed the importance of completing the square in algebra.

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Introduction

Completing the square is a powerful algebraic 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 provide a Q&A guide on completing the square, covering common questions and topics related to this technique.

Q: What is completing the square?

A: Completing the square is a method of solving quadratic equations of the form $ax^2 + bx + c = 0$ . It involves manipulating the equation to express it in the form $(x + \frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b}{2a})^2$ . This form is called a perfect square trinomial, and it can be easily solved by taking the square root of both sides.

Q: How do I know if an equation can be solved by completing the square?

A: An equation can be solved by completing the square if it is in the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. If the equation is in this form, you can try completing the square to solve it.

Q: What is the first step in completing the square?

A: The first step in completing the square is to move the constant term to the right side of the equation. This will give you an equation in the form $ax^2 + bx = c$ .

Q: How do I add and subtract the square of half the coefficient of the linear term?

A: To add and subtract the square of half the coefficient of the linear term, you need to find the value of half the coefficient of the linear term, square it, and then add and subtract it from the equation. For example, if the coefficient of the linear term is 24, you would add and subtract $(24/2)^2 = 144$ .

Q: What is the next step after adding and subtracting the square of half the coefficient of the linear term?

A: After adding and subtracting the square of half the coefficient of the linear term, you need to factor the perfect square trinomial. This will give you an equation in the form $(x + \frac{b}{2a})^2 = -\frac{c}{a} + (\frac{b}{2a})^2$ .

Q: How do I take the square root of both sides of the equation?

A: To take the square root of both sides of the equation, you need to isolate the square root term. This will give you an equation in the form $x + \frac{b}{2a} = \pm \sqrt{-\frac{c}{a} + (\frac{b}{2a})^2}$ .

Q: What is the final step in completing the square?

A: The final step in completing the square is to solve for x by subtracting $\frac{b}{2a}$ from both sides of the equation.

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 the linear term
Not factoring the perfect square trinomial
Not taking the square root of both sides of the equation
Not solving for x by subtracting $\frac{b}{2a}$ from both sides of the equation

Q: When should I use completing the square?

A: You should use completing the square when you are solving a quadratic equation of the form $ax^2 + bx + c = 0$ , where $a$ , $b$ , and $c$ are constants. Completing the square is a powerful technique that can be used to solve quadratic equations that are not easily solvable by other methods.

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

A: Completing the square has a wide range of real-world applications, including:

Physics: Completing the square is used to solve problems involving motion, energy, and momentum.
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 sorting and searching.

Conclusion

In conclusion, completing the square is a powerful algebraic 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 have provided a Q&A guide on completing the square, covering common questions and topics related to this technique. We hope that this guide has been helpful in understanding the concept of completing the square and how to apply it to solve quadratic equations.

Final Thoughts

Completing the square is a fundamental concept in algebra that has a wide range of applications in various fields. It is an important technique used to solve quadratic equations and is used in a variety of applications, including physics, engineering, and computer science. In this article, we have provided a Q&A guide on completing the square, covering common questions and topics related to this technique. We hope that this guide has been helpful in understanding the concept of completing the square and how to apply it to solve quadratic equations.