How Many More Unit Tiles Must Be Added To The Function $f(x)=x^2-6x+1$ In Order To Complete The Square?A. 1 B. 6 C. 8 D. 9

Feb 27, 2025 by ADMIN 130 views

**Completing the Square: A Step-by-Step Guide**

Introduction

Completing the square is a powerful technique used to rewrite quadratic expressions in a specific form, making it easier to solve equations and analyze functions. In this article, we will explore how to complete the square for the given function $f(x)=x^2-6x+1$ and determine how many more unit tiles must be added to complete the square.

What is Completing the Square?

Completing the square is a method of rewriting a quadratic expression in the form $(x-h)^2+k$, where $(x-h)^2$ is a perfect square trinomial and $k$ is a constant. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic expression.

The Process of Completing the Square

To complete the square for a quadratic expression in the form $ax^2+bx+c$, we follow these steps:

Ensure the coefficient of $x^2$ is 1: If the coefficient of $x^2$ is not 1, we need to factor it out. For example, if we have $2x^2+4x+1$, we can factor out 2 to get $2(x^2+2x+\frac{1}{2})$.
Move the constant term to the right-hand side: We move the constant term to the right-hand side of the equation by subtracting it from both sides. For example, if we have $x^2+4x+1$, we can move the constant term to the right-hand side by subtracting 1 from both sides, resulting in $x^2+4x=-1$.
Add and subtract $(\frac{b}{2})^2$ : We add and subtract $(\frac{b}{2})^2$ to the left-hand side of the equation, where $b$ is the coefficient of $x$ . This creates a perfect square trinomial. For example, if we have $x^2+4x=-1$, we can add and subtract $(\frac{4}{2})^2=4$ to the left-hand side, resulting in $x^2+4x+4-4=-1$.
Factor the perfect square trinomial: We factor the perfect square trinomial on the left-hand side of the equation. For example, if we have $x^2+4x+4-4=-1$, we can factor the perfect square trinomial to get $(x+2)^2-4=-1$.
Simplify the equation: We simplify the equation by combining like terms. For example, if we have $(x+2)^2-4=-1$, we can simplify the equation by adding 4 to both sides, resulting in $(x+2)^2=3$.

Completing the Square for the Given Function

Now that we have a good understanding of the process of completing the square, let's apply it to the given function $f(x)=x^2-6x+1$.

First, we need to ensure the coefficient of $x^2$ is 1. Since the coefficient of $x^2$ is already 1, we can move on to the next step.

Next, we move the constant term to the right-hand side by subtracting 1 from both sides, resulting in $x^2-6x=-1$.

Now, we add and subtract $(\frac{-6}{2})^2=9$ to the left-hand side of the equation, resulting in $x^2-6x+9-9=-1$.

We can factor the perfect square trinomial on the left-hand side of the equation to get $(x-3)^2-9=-1$.

Finally, we simplify the equation by combining like terms, resulting in $(x-3)^2=8$.

Determining the Number of Unit Tiles Needed

Now that we have completed the square for the given function, we can determine how many more unit tiles must be added to complete the square.

The completed square form of the function is $(x-3)^2=8$. To complete the square, we need to add 9 to the left-hand side of the equation, resulting in $(x-3)^2+9=8+9$.

Since we already have 9 on the left-hand side of the equation, we don't need to add any more unit tiles to complete the square.

Conclusion

In this article, we explored how to complete the square for the given function $f(x)=x^2-6x+1$. We applied the steps of completing the square to rewrite the function in the form $(x-h)^2+k$. We also determined how many more unit tiles must be added to complete the square, which is 0.

Final Answer

Introduction

Completing the square is a powerful technique used to rewrite quadratic expressions in a specific form, making it easier to solve equations and analyze functions. In this article, we will explore some common questions and answers related to completing the square.

Q: What is the purpose of completing the square?

A: The purpose of completing the square is to rewrite a quadratic expression in the form $(x-h)^2+k$, where $(x-h)^2$ is a perfect square trinomial and $k$ is a constant. This form is useful because it allows us to easily identify the vertex of the parabola represented by the quadratic expression.

Q: How do I know if a quadratic expression can be completed to a perfect square?

A: A quadratic expression can be completed to a perfect square if and only if the coefficient of the linear term is even. If the coefficient of the linear term is odd, then the quadratic expression cannot be completed to a perfect square.

Q: What is the difference between completing the square and factoring?

A: Completing the square and factoring are two different techniques used to rewrite quadratic expressions. Factoring involves expressing a quadratic expression as a product of two binomials, while completing the square involves rewriting a quadratic expression in the form $(x-h)^2+k$.

Q: Can I complete the square for a quadratic expression with a negative leading coefficient?

A: Yes, you can complete the square for a quadratic expression with a negative leading coefficient. However, you will need to take the square root of the negative number and then square it again to get the correct result.

Q: How do I determine the number of unit tiles needed to complete the square?

A: To determine the number of unit tiles needed to complete the square, you need to add and subtract $(\frac{b}{2})^2$ to the left-hand side of the equation, where $b$ is the coefficient of the linear term. The number of unit tiles needed is equal to the value that is added and subtracted.

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 $(x-h)^2=k$, you can easily identify the solutions to 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 ensuring the coefficient of the linear term is even
Not adding and subtracting the correct value
Not factoring the perfect square trinomial correctly
Not simplifying the equation correctly

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

A: To know if you have completed the square correctly, you need to check that the left-hand side of the equation is a perfect square trinomial and that the right-hand side is a constant.

Conclusion

In this article, we explored some common questions and answers related to completing the square. We discussed the purpose of completing the square, how to determine if a quadratic expression can be completed to a perfect square, and how to complete the square for a quadratic expression with a negative leading coefficient. We also discussed how to determine the number of unit tiles needed to complete the square and how to use completing the square to solve quadratic equations. Finally, we discussed some common mistakes to avoid when completing the square and how to know if you have completed the square correctly.

Final Answer

The final answer is that completing the square is a powerful technique used to rewrite quadratic expressions in a specific form, making it easier to solve equations and analyze functions. By following the steps outlined in this article, you can complete the square for any quadratic expression and determine the number of unit tiles needed to complete the square.