Factor The Expression Below.$x^2 - 16x + 64$A. $(x-8)(x-8$\] B. $(x-4)(x-16$\] C. $(x+4)(x+16$\] D. $(x+8)(x+8$\]

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 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 binomial expressions. This is done by finding the factors of the quadratic expression, which are the numbers or expressions that, when multiplied together, give the original quadratic expression.

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 quadratic expression into two pairs and then factoring out the greatest common factor (GCF) of each pair.

Step 1: Group the Terms

Group the terms of the quadratic expression into two pairs:

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

Step 2: Factor out the GCF

Factor out the GCF of each pair:

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

Step 3: Factor the Remaining Expression

Factor the remaining expression:

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

Step 4: Write the Final Factored Form

Write the final factored form:

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

Method 2: Factoring by Using the Perfect Square Trinomial Formula

Another method of factoring quadratic expressions is by using the perfect square trinomial formula. This involves recognizing that the quadratic expression is a perfect square trinomial and then factoring it accordingly.

Step 1: Identify the Perfect Square Trinomial

Identify the perfect square trinomial:

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

Step 2: Write the Final Factored Form

Write the final factored form:

$(x - 8)^2$

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 two methods: factoring by grouping and factoring by using the perfect square trinomial formula. The final factored form of the expression is $(x - 8)^2$.

Answer

The correct answer is:

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

Discussion

This problem is a great example of how factoring quadratic expressions can be used to simplify complex expressions and solve equations. The method of factoring by grouping is a useful technique to have in your toolkit, as it can be used to factor a wide range of quadratic expressions. Additionally, recognizing perfect square trinomials is an important skill to have, as it can save you time and effort when factoring quadratic expressions.

Related Topics

• Factoring quadratic expressions with two variables
• Factoring quadratic expressions with complex coefficients
• Solving quadratic equations using factoring

Practice Problems

• Factor the expression $x^2 + 14x + 49$
• Factor the expression $x^2 - 20x + 100$
• Factor the expression $x^2 + 12x + 36$

Glossary

• Factoring: The process of expressing a quadratic expression as a product of two or more binomial expressions.
• Greatest Common Factor (GCF): The largest number or expression that divides each term of a quadratic expression without leaving a remainder.
• Perfect Square Trinomial: A quadratic expression that can be factored into the square of a binomial expression.
  Factoring Quadratic Expressions: A Q&A Guide =====================================================

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 answer some of the most frequently asked questions about factoring quadratic expressions.

Q: What is factoring?

A: Factoring is the process of expressing a quadratic expression as a product of two or more binomial expressions. This is done by finding the factors of the quadratic expression, which are the numbers or expressions that, when multiplied together, give the original quadratic 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 using the perfect square trinomial formula
• Factoring by using the difference of squares formula
• Factoring by using the sum and difference formulas

Q: How do I factor a quadratic expression using the factoring by grouping method?

A: To factor a quadratic expression using the factoring by grouping method, follow these steps:

1. Group the terms of the quadratic expression into two pairs.
2. Factor out the greatest common factor (GCF) of each pair.
3. Factor the remaining expression.

Q: How do I factor a quadratic expression using the perfect square trinomial formula?

A: To factor a quadratic expression using the perfect square trinomial formula, follow these steps:

1. Identify the perfect square trinomial.
2. Write the final factored form.

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 expression.

Q: How do I identify a perfect square trinomial?

A: To identify a perfect square trinomial, look for the following characteristics:

• The first and last terms are perfect squares.
• The middle term is twice the product of the square roots of the first and last terms.

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 identifying the perfect square trinomial.
• Not using the correct formula for factoring.

Q: How do I know which method to use when factoring a quadratic expression?

A: To determine which method to use when factoring a quadratic expression, follow these steps:

1. Look for the greatest common factor (GCF) of each pair of terms.
2. Check if the quadratic expression is a perfect square trinomial.
3. Use the difference of squares formula if the quadratic expression is in the form of (x - a)(x - b).
4. Use the sum and difference formulas if the quadratic expression is in the form of (x + a)(x + b).

Q: Can I factor a quadratic expression with complex coefficients?

A: Yes, you can factor a quadratic expression with complex coefficients. However, you will need to use the complex conjugate to factor the expression.

Q: Can I factor a quadratic expression with two variables?

A: Yes, you can factor a quadratic expression with two variables. However, you will need to use the method of substitution to factor the expression.

Conclusion

In this article, we have answered some of the most frequently asked questions about factoring quadratic expressions. We have covered the different methods of factoring, including factoring by grouping, factoring by using the perfect square trinomial formula, and factoring by using the difference of squares formula. We have also discussed some common mistakes to avoid when factoring quadratic expressions and provided tips on how to determine which method to use when factoring a quadratic expression.

Practice Problems

• Factor the expression $x^2 + 14x + 49$
• Factor the expression $x^2 - 20x + 100$
• Factor the expression $x^2 + 12x + 36$

Glossary

• Factoring: The process of expressing a quadratic expression as a product of two or more binomial expressions.
• Greatest Common Factor (GCF): The largest number or expression that divides each term of a quadratic expression without leaving a remainder.
• Perfect Square Trinomial: A quadratic expression that can be factored into the square of a binomial expression.
• Complex Conjugate: A complex number that is the conjugate of another complex number.
• Method of Substitution: A method of factoring quadratic expressions with two variables.