Factor The Expression Below. X 2 − 16 X + 64 X^2 - 16x + 64 X 2 − 16 X + 64 A. { (x-8)(x-8)$}$B. { (x+8)(x+8)$}$C. { (x+4)(x+16)$}$D. { (x-4)(x-16)$}$

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we will focus on factoring the expression $x^2 - 16x + 64$ and explore the different methods and techniques used to factor quadratic expressions.

What is Factoring?

Factoring is the process of expressing a quadratic expression as a product of two or more binomials. This involves finding the factors of the quadratic expression that, when multiplied together, result in the original expression. Factoring quadratic expressions is an essential skill in algebra, as it allows us to simplify complex expressions, solve equations, and graph functions.

The Expression to be Factored

The expression we will be factoring is $x^2 - 16x + 64$. This is a quadratic expression in the form of $ax^2 + bx + c$, where $a = 1$, $b = -16$, and $c = 64$.

Method 1: Factoring by Grouping

One method of factoring quadratic expressions is by grouping. This involves grouping the terms of the expression into pairs and then factoring out the greatest common factor (GCF) of each pair.

Step 1: Group the Terms

The first step in factoring by grouping is to group the terms of the expression into pairs. In this case, we can group the terms as follows:

$x^2 - 16x + 64 = (x^2 - 16x) + 64$

Step 2: Factor Out the GCF

The next step is to factor out the greatest common factor (GCF) of each pair. In this case, the GCF of the first pair is $x$, and the GCF of the second pair is $64$.

$(x^2 - 16x) + 64 = x(x - 16) + 64$

Step 3: Factor Out the Common Factor

The final step is to factor out the common factor of the two terms. In this case, the common factor is $1$, but we can rewrite the expression as follows:

$x(x - 16) + 64 = (x - 8)(x - 8)$

Method 2: Factoring by Completing the Square

Another method of factoring quadratic expressions is by completing the square. This involves rewriting the quadratic expression in the form of $(x + a)^2 + b$, where $a$ and $b$ are constants.

Step 1: Rewrite the Expression

The first step in factoring by completing the square is to rewrite the quadratic expression in the form of $(x + a)^2 + b$. In this case, we can rewrite the expression as follows:

$x^2 - 16x + 64 = (x - 8)^2 + 0$

Step 2: Factor Out the Square

The next step is to factor out the square. In this case, the square is $(x - 8)$.

$(x - 8)^2 + 0 = (x - 8)(x - 8)$

Method 3: Factoring by Using the Formula

Another method of factoring quadratic expressions is by using the formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$. This involves using the quadratic formula to find the roots of the quadratic expression.

Step 1: Plug in the Values

The first step in factoring by using the formula is to plug in the values of $a$, $b$, and $c$ into the quadratic formula. In this case, we have $a = 1$, $b = -16$, and $c = 64$.

$x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4(1)(64)}}{2(1)}$

Step 2: Simplify the Expression

The next step is to simplify the expression. In this case, we can simplify the expression as follows:

$x = \frac{16 \pm \sqrt{256 - 256}}{2}$

$x = \frac{16 \pm \sqrt{0}}{2}$

$x = \frac{16}{2}$

$x = 8$

Step 3: Factor Out the Root

The final step is to factor out the root. In this case, the root is $x - 8$.

$x = 8 = (x - 8)(x - 8)$

Conclusion

In this article, we have explored the different methods and techniques used to factor quadratic expressions. We have factored the expression $x^2 - 16x + 64$ using three different methods: factoring by grouping, factoring by completing the square, and factoring by using the formula. Each method has its own advantages and disadvantages, and the choice of method depends on the specific expression being factored.

Answer

The correct answer is:

A. {(x-8)(x-8)$}$

$$

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that allows us to simplify complex expressions and solve equations. In this article, we will provide a Q&A guide to help you understand the different methods and techniques used to factor quadratic expressions.

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two or more binomials. This involves finding the factors of the quadratic expression that, when multiplied together, result in the original expression.

Q: What are the different methods of factoring quadratic expressions?

A: There are several methods of factoring quadratic expressions, including:

• Factoring by grouping
• Factoring by completing the square
• Factoring by using the formula
• Factoring by using the quadratic formula

Q: What is factoring by grouping?

A: Factoring by grouping involves grouping the terms of the expression into pairs and then factoring out the greatest common factor (GCF) of each pair.

Q: What is factoring by completing the square?

A: Factoring by completing the square involves rewriting the quadratic expression in the form of $(x + a)^2 + b$, where $a$ and $b$ are constants.

Q: What is factoring by using the formula?

A: Factoring by using the formula involves using the quadratic formula to find the roots of the quadratic expression.

Q: What is the quadratic formula?

A: The quadratic formula is $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the quadratic expression.

Q: How do I choose the correct method of factoring?

A: The choice of method depends on the specific expression being factored. You should try each method and see which one works best.

Q: What are some common mistakes to avoid when factoring quadratic expressions?

A: Some common mistakes to avoid when factoring quadratic expressions include:

• Not factoring out the greatest common factor (GCF) of each pair
• Not rewriting the expression in the correct form
• Not using the correct formula
• Not checking the solutions

Q: How do I check the solutions?

A: To check the solutions, you should plug the values back into the original expression and simplify. If the expression equals zero, then the solution is correct.

Q: What are some real-world applications of factoring quadratic expressions?

A: Factoring quadratic expressions has many real-world applications, including:

• Solving equations in physics and engineering
• Modeling population growth and decay
• Analyzing data in statistics and economics
• Solving optimization problems in business and finance

Conclusion

In this article, we have provided a Q&A guide to help you understand the different methods and techniques used to factor quadratic expressions. We have also discussed some common mistakes to avoid and provided some real-world applications of factoring quadratic expressions.

Frequently Asked Questions

• Q: What is the difference between factoring and simplifying? A: Factoring involves expressing an expression as a product of two or more binomials, while simplifying involves combining like terms.
• Q: Can I factor a quadratic expression with a negative coefficient? A: Yes, you can factor a quadratic expression with a negative coefficient by using the same methods as before.
• Q: How do I factor a quadratic expression with a variable coefficient? A: You can factor a quadratic expression with a variable coefficient by using the same methods as before, but you may need to use a different formula or technique.

Additional Resources

• Factoring Quadratic Expressions Worksheet: A worksheet with practice problems to help you master factoring quadratic expressions.
• Factoring Quadratic Expressions Video: A video tutorial that explains the different methods and techniques used to factor quadratic expressions.
• Factoring Quadratic Expressions Online Calculator: An online calculator that can help you factor quadratic expressions quickly and easily.