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

Introduction to 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 particularly useful in solving quadratic equations and in graphing quadratic functions. The process of completing the square involves manipulating the quadratic expression to create a perfect square trinomial, which can then be factored into the square of a binomial.

The Quadratic Expression

The given quadratic expression is $x^2 - x$. To complete the square, we need to add a constant term to this expression such that it becomes a perfect square trinomial.

The Formula for Completing the Square

The formula for completing the square is based on the concept of the square of a binomial. The square of a binomial $(x + a)$ is given by $(x + a)^2 = x^2 + 2ax + a^2$. To complete the square for the quadratic expression $x^2 - x$, we need to find the value of $a$ such that the expression becomes a perfect square trinomial.

Finding the Value of $a$

To find the value of $a$, we need to compare the given quadratic expression $x^2 - x$ with the formula for the square of a binomial. We can see that the coefficient of $x$ in the given expression is $-1$, which is half of the coefficient of $x$ in the formula for the square of a binomial. Therefore, we can conclude that $a = -\frac{1}{2}$.

Completing the Square

Now that we have found the value of $a$, we can complete the square by adding the square of $a$ to the given quadratic expression. The square of $a$ is given by $a^2 = \left(-\frac{1}{2}\right)^2 = \frac{1}{4}$. Therefore, we need to add $\frac{1}{4}$ to the given quadratic expression to complete the square.

The Completed Square

The completed square is given by $x^2 - x + \frac{1}{4}$. This expression can be rewritten as $\left(x - \frac{1}{2}\right)^2$. Therefore, the number that must be added to the given quadratic expression to complete the square is $\frac{1}{4}$.

Conclusion

In conclusion, the number that must be added to the expression $x^2 - x$ to complete the square is $\frac{1}{4}$. This is based on the formula for completing the square, which involves adding the square of half the coefficient of $x$ to the quadratic expression. The completed square is given by $\left(x - \frac{1}{2}\right)^2$, which is a perfect square trinomial.

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 useful? Completing the square is useful in solving quadratic equations and in graphing quadratic functions.
• How do I complete the square? To complete the square, you need to add the square of half the coefficient of $x$ to the quadratic expression.

Final Answer

The final answer is $\boxed{\frac{1}{4}}$.

Introduction

Completing the square is a powerful mathematical technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This method is particularly useful in solving quadratic equations and in graphing quadratic functions. In this article, we will provide a comprehensive guide to completing the square, along with frequently asked questions and answers.

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. This involves adding a constant term to the quadratic expression such that it becomes a perfect square trinomial.

Why is Completing the Square Useful?

Completing the square is useful in solving quadratic equations and in graphing quadratic functions. It allows us to rewrite the quadratic expression in a form that is easier to work with, making it easier to solve for the roots of the equation.

How Do I Complete the Square?

To complete the square, you need to follow these steps:

1. Identify the quadratic expression: The quadratic expression is usually in the form of $ax^2 + bx + c$.
2. Find the value of $a$: The value of $a$ is the coefficient of $x^2$ in the quadratic expression.
3. Find the value of $b$: The value of $b$ is the coefficient of $x$ in the quadratic expression.
4. Add the square of half the value of $b$: Add $\left(\frac{b}{2}\right)^2$ to the quadratic expression.
5. Simplify the expression: Simplify the resulting expression to obtain the completed square.

Examples of Completing the Square

Example 1

Complete the square for the quadratic expression $x^2 + 4x$.

• Identify the quadratic expression: $x^2 + 4x$
• Find the value of $a$: $a = 1$
• Find the value of $b$: $b = 4$
• Add the square of half the value of $b$: $\left(\frac{4}{2}\right)^2 = 4$
• Simplify the expression: $x^2 + 4x + 4 = (x + 2)^2$

Example 2

Complete the square for the quadratic expression $x^2 - 6x$.

• Identify the quadratic expression: $x^2 - 6x$
• Find the value of $a$: $a = 1$
• Find the value of $b$: $b = -6$
• Add the square of half the value of $b$: $\left(\frac{-6}{2}\right)^2 = 9$
• Simplify the expression: $x^2 - 6x + 9 = (x - 3)^2$

Frequently Asked Questions

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.

Q: Why is completing the square useful?

A: Completing the square is useful in solving quadratic equations and in graphing quadratic functions.

Q: How do I complete the square?

A: To complete the square, you need to add the square of half the coefficient of $x$ to the quadratic expression.

Q: What is the formula for completing the square?

A: The formula for completing the square is $(x + a)^2 = x^2 + 2ax + a^2$.

Q: How do I find the value of $a$?

A: To find the value of $a$, you need to identify the coefficient of $x^2$ in the quadratic expression.

Q: How do I find the value of $b$?

A: To find the value of $b$, you need to identify the coefficient of $x$ in the quadratic expression.

Q: What is the completed square?

A: The completed square is the resulting expression after adding the square of half the coefficient of $x$ to the quadratic expression.

Conclusion

In conclusion, completing the square is a powerful mathematical technique used to rewrite a quadratic expression in the form of a perfect square trinomial. This method is particularly useful in solving quadratic equations and in graphing quadratic functions. By following the steps outlined in this article, you can complete the square for any quadratic expression.

Final Answer

The final answer is $\boxed{\frac{1}{4}}$.