Which Of The Following Completes The Perfect Square Trinomial For The Expression $x^2 - 6x$?A. $9$B. [tex$-36$[/tex]C. [tex 99 99 99 D . [ T E X D. [tex D . [ T E X 36 36 36 [/tex]

Introduction

In algebra, a perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It is a crucial concept in mathematics, particularly in algebra and calculus. In this article, we will explore the concept of perfect square trinomial and provide a step-by-step guide on how to complete the perfect square trinomial for a given expression.

What is a Perfect Square Trinomial?

A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form:

$$a^2 + 2ab + b^2$$

where $a$ and $b$ are constants. The expression $a^2 + 2ab + b^2$ is a perfect square trinomial because it can be factored into the square of a binomial:

$$(a + b)^2$$

Completing the Perfect Square Trinomial

To complete the perfect square trinomial for a given expression, we need to find the values of $a$ and $b$ that make the expression a perfect square trinomial. The general form of a perfect square trinomial is:

$$a^2 + 2ab + b^2$$

We can complete the perfect square trinomial by finding the values of $a$ and $b$ that satisfy the following equation:

$$a^2 + 2ab + b^2 = x^2 - 6x$$

Step 1: Identify the Values of $a$ and $b$

To complete the perfect square trinomial, we need to identify the values of $a$ and $b$ that satisfy the equation:

$$a^2 + 2ab + b^2 = x^2 - 6x$$

We can start by identifying the values of $a$ and $b$ that make the expression $x^2 - 6x$ a perfect square trinomial.

Step 2: Factor the Expression

To factor the expression $x^2 - 6x$, we need to find the values of $a$ and $b$ that make the expression a perfect square trinomial. We can start by factoring the expression:

$$x^2 - 6x = (x - 3)^2 - 9$$

Step 3: Identify the Values of $a$ and $b$

Now that we have factored the expression, we can identify the values of $a$ and $b$ that make the expression a perfect square trinomial. We can see that:

$$a = x - 3$$

and

$$b = 3$$

Step 4: Complete the Perfect Square Trinomial

Now that we have identified the values of $a$ and $b$, we can complete the perfect square trinomial. We can see that:

$$a^2 + 2ab + b^2 = (x - 3)^2 + 2(x - 3)(3) + 3^2$$

Simplifying the expression, we get:

$$(x - 3)^2 + 6(x - 3) + 9$$

Conclusion

In this article, we have explored the concept of perfect square trinomial and provided a step-by-step guide on how to complete the perfect square trinomial for a given expression. We have identified the values of $a$ and $b$ that make the expression $x^2 - 6x$ a perfect square trinomial and completed the perfect square trinomial.

Which of the Following Completes the Perfect Square Trinomial for the Expression $x^2 - 6x$?

A. $9$ B. $-36$ C. $99$ D. $36$

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Q&A: Perfect Square Trinomial

Q: What is a perfect square trinomial?

A: A perfect square trinomial is a quadratic expression that can be factored into the square of a binomial. It has the form:

$$a^2 + 2ab + b^2$$

where $a$ and $b$ are constants.

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, you need to look for an expression that can be factored into the square of a binomial. You can do this by checking if the expression has the form:

$$a^2 + 2ab + b^2$$

Q: How do I complete the perfect square trinomial for a given expression?

A: To complete the perfect square trinomial for a given expression, you need to follow these steps:

1. Identify the values of $a$ and $b$ that make the expression a perfect square trinomial.
2. Factor the expression using the values of $a$ and $b$.
3. Complete the perfect square trinomial by adding the square of the binomial to the expression.

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

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

• Not identifying the values of $a$ and $b$ correctly.
• Not factoring the expression correctly.
• Not adding the square of the binomial to the expression.

Q: How do I check if an expression is a perfect square trinomial?

A: To check if an expression is a perfect square trinomial, you can use the following steps:

1. Check if the expression has the form:

$$a^2 + 2ab + b^2$$

2. Factor the expression using the values of $a$ and $b$.
3. Check if the expression can be factored into the square of a binomial.

Q: What are some real-world applications of perfect square trinomials?

A: Perfect square trinomials have many real-world applications, including:

• Algebra and calculus
• Physics and engineering
• Computer science and programming

Q: How do I use perfect square trinomials in algebra and calculus?

A: Perfect square trinomials are used extensively in algebra and calculus to solve quadratic equations and optimize functions. They are also used to find the maximum and minimum values of functions.

Q: What are some common types of perfect square trinomials?

A: Some common types of perfect square trinomials include:

• Monic perfect square trinomials: These are perfect square trinomials where the leading coefficient is 1.
• Non-monic perfect square trinomials: These are perfect square trinomials where the leading coefficient is not 1.

Q: How do I factor a perfect square trinomial?

A: To factor a perfect square trinomial, you need to follow these steps:

1. Identify the values of $a$ and $b$ that make the expression a perfect square trinomial.
2. Factor the expression using the values of $a$ and $b$.
3. Write the expression as the square of a binomial.

Q: What are some common mistakes to avoid when factoring a perfect square trinomial?

A: Some common mistakes to avoid when factoring a perfect square trinomial include:

• Not identifying the values of $a$ and $b$ correctly.
• Not factoring the expression correctly.
• Not writing the expression as the square of a binomial.

Conclusion

In this article, we have provided a comprehensive guide to perfect square trinomials, including a step-by-step guide on how to complete the perfect square trinomial for a given expression. We have also answered some common questions about perfect square trinomials, including how to identify a perfect square trinomial, how to complete the perfect square trinomial, and how to factor a perfect square trinomial.