What Number Must Be Added To The Expression Below To Complete The Square?${ X^2 + X }$A. 2 B. 1 C. { \frac{1}{2}$}$ D. { \frac{1}{4}$}$

What Number Must Be Added to the Expression Below to Complete the Square?

Understanding the Concept of Completing the Square

Completing the square is a mathematical technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This method is essential in algebra, as it allows us to solve quadratic equations and analyze their properties. In this article, we will focus on the process of completing the square and determine the number that must be added to the given expression to achieve this.

The Formula for Completing the Square

The general formula for completing the square is:

${ x^2 + bx + \left( \frac{b}{2} \right)^2 }$

where ${ b }$ is the coefficient of the linear term in the quadratic expression. To complete the square, we need to add the square of half the coefficient of the linear term to the expression.

Applying the Formula to the Given Expression

In the given expression, ${ x^2 + x }$, the coefficient of the linear term is 1. To complete the square, we need to add the square of half of this coefficient, which is:

${ \left( \frac{1}{2} \right)^2 = \frac{1}{4} }$

Therefore, the number that must be added to the expression to complete the square is ${ \frac{1}{4} }$.

Why is Completing the Square Important?

Completing the square is an essential technique in algebra, as it allows us to solve quadratic equations and analyze their properties. By rewriting a quadratic expression in the form of a perfect square trinomial, we can easily identify the vertex of the parabola and determine the maximum or minimum value of the function.

Real-World Applications of Completing the Square

Completing the square has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, completing the square is used to describe the motion of objects under the influence of gravity. In engineering, it is used to design and optimize systems, such as bridges and buildings. In economics, it is used to model and analyze economic systems, such as supply and demand curves.

Conclusion

In conclusion, completing the square is a powerful technique in algebra that allows us to rewrite a quadratic expression in the form of a perfect square trinomial. By adding the square of half the coefficient of the linear term to the expression, we can easily identify the vertex of the parabola and determine the maximum or minimum value of the function. In this article, we have determined that the number that must be added to the expression ${ x^2 + x }$ to complete the square is ${ \frac{1}{4} }$.

Frequently Asked Questions

• What is completing the square? Completing the square is a mathematical technique used to rewrite a quadratic expression in the form of a perfect square trinomial.
• Why is completing the square important? Completing the square is essential in algebra, as it allows us to solve quadratic equations and analyze their properties.
• What is the formula for completing the square? The general formula for completing the square is: ${ x^2 + bx + \left( \frac{b}{2} \right)^2 }$
• What number must be added to the expression ${ x^2 + x }$ to complete the square? The number that must be added to the expression ${ x^2 + x }$ to complete the square is ${ \frac{1}{4} }$.

References

• Algebra and Trigonometry by Michael Sullivan
• Calculus by Michael Spivak
• Mathematics for Engineers and Scientists by Donald R. Hill

Glossary

• Quadratic expression: A polynomial expression of degree two, in the form of ${ ax^2 + bx + c }$.
• Perfect square trinomial: A quadratic expression that can be rewritten in the form of ${ (x + a)^2 }$.
• Vertex: The point on the parabola where the function reaches its maximum or minimum value.
  Completing the Square: A Q&A Article

Introduction

Completing the square is a fundamental concept in algebra that allows us to rewrite a quadratic expression in the form of a perfect square trinomial. In this article, we will answer some of the most frequently asked questions about completing the square, providing a deeper understanding of this essential technique.

Q: What is completing the square?

A: Completing the square is a mathematical technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This involves adding a constant term to the expression to make it a perfect square.

Q: Why is completing the square important?

A: Completing the square is essential in algebra, as it allows us to solve quadratic equations and analyze their properties. By rewriting a quadratic expression in the form of a perfect square trinomial, we can easily identify the vertex of the parabola and determine the maximum or minimum value of the function.

Q: What is the formula for completing the square?

A: The general formula for completing the square is:

${ x^2 + bx + \left( \frac{b}{2} \right)^2 }$

where ${ b }$ is the coefficient of the linear term in the quadratic expression.

Q: How do I complete the square for a quadratic expression?

A: To complete the square for a quadratic expression, follow these steps:

1. Identify the coefficient of the linear term, ${ b }$.
2. Add the square of half the coefficient of the linear term to the expression.
3. Simplify the expression to obtain the perfect square trinomial.

Q: What number must be added to the expression ${ x^2 + x }$ to complete the square?

A: The number that must be added to the expression ${ x^2 + x }$ to complete the square is ${ \frac{1}{4} }$.

Q: Can I complete the square for any quadratic expression?

A: Yes, you can complete the square for any quadratic expression. However, the expression must be in the form of ${ ax^2 + bx + c }$, where ${ a }$, ${ b }$, and ${ c }$ are constants.

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

A: Completing the square has numerous real-world applications in fields such as physics, engineering, and economics. For example, in physics, completing the square is used to describe the motion of objects under the influence of gravity. In engineering, it is used to design and optimize systems, such as bridges and buildings. In economics, it is used to model and analyze economic systems, such as supply and demand curves.

Q: How do I determine the vertex of a parabola using completing the square?

A: To determine the vertex of a parabola using completing the square, follow these steps:

1. Complete the square for the quadratic expression.
2. Identify the perfect square trinomial.
3. Determine the vertex of the parabola by identifying the value of ${ x }$ that makes the expression equal to zero.

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

A: Yes, you can use completing the square to solve quadratic equations. By rewriting the quadratic equation in the form of a perfect square trinomial, you can easily identify the solutions to the equation.

Conclusion

In conclusion, completing the square is a powerful technique in algebra that allows us to rewrite a quadratic expression in the form of a perfect square trinomial. By answering some of the most frequently asked questions about completing the square, we have provided a deeper understanding of this essential technique. Whether you are a student or a professional, completing the square is an essential tool to have in your mathematical toolkit.

Frequently Asked Questions

• What is completing the square? Completing the square is a mathematical technique used to rewrite a quadratic expression in the form of a perfect square trinomial.
• Why is completing the square important? Completing the square is essential in algebra, as it allows us to solve quadratic equations and analyze their properties.
• What is the formula for completing the square? The general formula for completing the square is: ${ x^2 + bx + \left( \frac{b}{2} \right)^2 }$
• How do I complete the square for a quadratic expression? To complete the square for a quadratic expression, follow these steps: 1. Identify the coefficient of the linear term, ${ b }$. 2. Add the square of half the coefficient of the linear term to the expression. 3. Simplify the expression to obtain the perfect square trinomial.

References

• Algebra and Trigonometry by Michael Sullivan
• Calculus by Michael Spivak
• Mathematics for Engineers and Scientists by Donald R. Hill

Glossary

• Quadratic expression: A polynomial expression of degree two, in the form of ${ ax^2 + bx + c }$.
• Perfect square trinomial: A quadratic expression that can be rewritten in the form of ${ (x + a)^2 }$.
• Vertex: The point on the parabola where the function reaches its maximum or minimum value.