Identify The Factors Of $x^2 + 36y$.A. PrimeB. $(x + 6y)(x - 6y)$C. \$(x + 6y)(x + 6y)$[/tex\]D. $(x - 6y)(x - 6y)$

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. In this article, we will explore the factors of the quadratic expression $x^2 + 36y$ and provide a step-by-step guide on how to factor it.

Understanding Quadratic Expressions

A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants. In the case of the expression $x^2 + 36y$, we can see that it is a quadratic expression with a leading coefficient of 1 and a constant term of 36y.

Factoring Quadratic Expressions

To factor a quadratic expression, we need to find two binomials whose product is equal to the original expression. The general form of a factored quadratic expression is $(x + m)(x + n)$, where $m$ and $n$ are constants. In the case of the expression $x^2 + 36y$, we can see that it can be factored as $(x + 6y)(x - 6y)$.

Why is the Correct Answer (x + 6y)(x - 6y)?

The correct answer is $(x + 6y)(x - 6y)$ because it satisfies the following conditions:

• The product of the two binomials is equal to the original expression: $(x + 6y)(x - 6y) = x^2 - 36y^2$
• The two binomials have a common factor of $x$: $(x + 6y)$ and $(x - 6y)$ both have an $x$ term
• The two binomials have a difference of squares: $(x + 6y)$ and $(x - 6y)$ both have a difference of squares, which is a common factorization technique

Why are the Other Options Incorrect?

The other options are incorrect because they do not satisfy the conditions mentioned above. For example:

• Option A, $(x + 6y)(x - 6y)$, is not a correct factorization of the expression $x^2 + 36y$.
• Option C, $(x + 6y)(x + 6y)$, is not a correct factorization of the expression $x^2 + 36y$ because it does not have a difference of squares.
• Option D, $(x - 6y)(x - 6y)$, is not a correct factorization of the expression $x^2 + 36y$ because it does not have a difference of squares.

Conclusion

In conclusion, the correct factorization of the quadratic expression $x^2 + 36y$ is $(x + 6y)(x - 6y)$. This is because it satisfies the conditions of having a product equal to the original expression, a common factor of $x$, and a difference of squares. The other options are incorrect because they do not satisfy these conditions.

Tips and Tricks

Here are some tips and tricks to help you factor quadratic expressions:

• Look for a difference of squares: If the expression can be written as a difference of squares, it can be factored as $(x + m)(x - m)$.
• Look for a common factor: If the expression has a common factor, it can be factored out.
• Use the FOIL method: The FOIL method is a technique for factoring quadratic expressions by multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms.

Examples and Practice Problems

Here are some examples and practice problems to help you practice factoring quadratic expressions:

• Factor the expression $x^2 + 12y$: $(x + 6y)(x - 6y)$
• Factor the expression $x^2 - 16y^2$: $(x + 4y)(x - 4y)$
• Factor the expression $x^2 + 10x + 25$: $(x + 5)(x + 5)$

Conclusion

Introduction

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. In this article, we will provide a Q&A guide to help you understand the concept of factoring quadratic expressions and how to apply it to solve problems.

Q: What is a quadratic expression?

A: A quadratic expression is a polynomial of degree two, which means it has a highest power of two. The general form of a quadratic expression is $ax^2 + bx + c$, where $a$, $b$, and $c$ are constants.

Q: What is factoring a quadratic expression?

A: Factoring a quadratic expression means expressing it as a product of two binomials. The general form of a factored quadratic expression is $(x + m)(x + n)$, where $m$ and $n$ are constants.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two binomials whose product is equal to the original expression. Here are the steps to follow:

1. Look for a difference of squares: If the expression can be written as a difference of squares, it can be factored as $(x + m)(x - m)$.
2. Look for a common factor: If the expression has a common factor, it can be factored out.
3. Use the FOIL method: The FOIL method is a technique for factoring quadratic expressions by multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms.

Q: What is the difference of squares?

A: The difference of squares is a technique for factoring quadratic expressions that can be written as a difference of squares. The general form of a difference of squares is $(x + m)(x - m)$, where $m$ is a constant.

Q: How do I use the FOIL method?

A: The FOIL method is a technique for factoring quadratic expressions by multiplying the first terms, then the outer terms, then the inner terms, and finally the last terms. Here are the steps to follow:

1. Multiply the first terms: $(x + m)(x + n) = x^2 + mx + nx$
2. Multiply the outer terms: $x^2 + mx + nx = x^2 + (m + n)x$
3. Multiply the inner terms: $x^2 + (m + n)x = x^2 + (m + n)x + mn$
4. Multiply the last terms: $x^2 + (m + n)x + mn = x^2 + (m + n)x + mn$

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

A: Here are some common mistakes to avoid when factoring quadratic expressions:

• Not looking for a difference of squares
• Not looking for a common factor
• Not using the FOIL method correctly
• Not checking the product of the two binomials

Q: How do I check if the product of the two binomials is correct?

A: To check if the product of the two binomials is correct, you need to multiply the two binomials and see if the result is equal to the original expression. Here are the steps to follow:

1. Multiply the two binomials: $(x + m)(x + n) = x^2 + mx + nx$
2. Simplify the expression: $x^2 + mx + nx = x^2 + (m + n)x + mn$
3. Check if the result is equal to the original expression: $x^2 + (m + n)x + mn = x^2 + bx + c$

Conclusion

Factoring quadratic expressions is a fundamental concept in algebra that helps us simplify complex equations and solve problems more efficiently. In this article, we provided a Q&A guide to help you understand the concept of factoring quadratic expressions and how to apply it to solve problems. We also discussed some common mistakes to avoid and how to check if the product of the two binomials is correct.