Write The Expression As A Product: 64 − ( B + 1 ) 2 64 - (b+1)^2 64 − ( B + 1 ) 2

Introduction

In algebra, it is often necessary to express an expression as a product, which can be a challenging task. One common method for doing this is by using the difference of squares formula. In this article, we will explore how to write the expression $64 - (b+1)^2$ as a product.

Understanding the Expression

The given expression is $64 - (b+1)^2$. This expression consists of two terms: a constant term $64$ and a squared term $(b+1)^2$. To write this expression as a product, we need to find a way to factorize the squared term.

Factoring the Squared Term

The squared term $(b+1)^2$ can be factored using the formula $(a+b)^2 = a^2 + 2ab + b^2$. In this case, we have $(b+1)^2 = b^2 + 2b(1) + 1^2$. Therefore, we can write the squared term as $(b+1)^2 = (b+1)(b+1)$.

Using the Difference of Squares Formula

The difference of squares formula states that $a^2 - b^2 = (a+b)(a-b)$. We can use this formula to write the expression $64 - (b+1)^2$ as a product. Let's start by rewriting the expression as $64 - (b+1)^2 = 64 - (b+1)(b+1)$.

Finding the Product

Now, we need to find a way to factorize the expression $64 - (b+1)(b+1)$. We can do this by using the difference of squares formula. Let's assume that $a^2 - b^2 = (a+b)(a-b)$. In this case, we have $64 - (b+1)(b+1) = a^2 - b^2$. Therefore, we can write the expression as $(a+b)(a-b)$.

Solving for a and b

Now, we need to solve for $a$ and $b$. We know that $a^2 - b^2 = 64 - (b+1)(b+1)$. Let's start by expanding the squared term: $(b+1)(b+1) = b^2 + 2b(1) + 1^2 = b^2 + 2b + 1$. Therefore, we have $a^2 - b^2 = 64 - (b^2 + 2b + 1)$.

Simplifying the Expression

Now, we can simplify the expression by combining like terms: $a^2 - b^2 = 64 - b^2 - 2b - 1 = 63 - b^2 - 2b$. Therefore, we have $a^2 - b^2 = 63 - b^2 - 2b$.

Factoring the Expression

Introduction

In our previous article, we explored how to write the expression $64 - (b+1)^2$ as a product. We used the difference of squares formula to factorize the squared term and simplify the expression. In this article, we will answer some common questions related to this topic.

Q: What is the difference of squares formula?

A: The difference of squares formula is a mathematical formula that states $a^2 - b^2 = (a+b)(a-b)$. This formula can be used to factorize the difference of two squares.

Q: How do I apply the difference of squares formula?

A: To apply the difference of squares formula, you need to identify the two squares in the expression. In the case of $64 - (b+1)^2$, the two squares are $64$ and $(b+1)^2$. You can then use the formula to factorize the expression.

Q: What if the expression is not a perfect square trinomial?

A: If the expression is not a perfect square trinomial, you may need to use other factoring techniques, such as grouping or factoring by grouping. However, in the case of $64 - (b+1)^2$, we were able to factorize the expression using the difference of squares formula.

Q: Can I use the difference of squares formula to factorize any expression?

A: No, the difference of squares formula can only be used to factorize expressions that are in the form of $a^2 - b^2$. If the expression is not in this form, you may need to use other factoring techniques.

Q: What are some common mistakes to avoid when using the difference of squares formula?

A: Some common mistakes to avoid when using the difference of squares formula include:

• Not identifying the two squares in the expression
• Not using the correct formula
• Not simplifying the expression correctly
• Not checking for any common factors

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:

• Expand the expression
• Check if the expanded expression is a perfect square trinomial
• If it is, then the original expression is a perfect square trinomial

Q: Can I use the difference of squares formula to solve equations?

A: Yes, the difference of squares formula can be used to solve equations. However, you need to be careful when using this formula, as it can lead to extraneous solutions.

Conclusion

In this article, we answered some common questions related to writing the expression $64 - (b+1)^2$ as a product. We also discussed the difference of squares formula and how to apply it to factorize expressions. We hope that this article has been helpful in understanding this topic.

Additional Resources

If you want to learn more about writing expressions as products, we recommend checking out the following resources:

• Khan Academy: Writing expressions as products
• Mathway: Writing expressions as products
• Wolfram Alpha: Writing expressions as products

Practice Problems

If you want to practice writing expressions as products, we recommend trying the following problems:

• Write the expression $25 - (x+2)^2$ as a product.
• Write the expression $36 - (y-3)^2$ as a product.
• Write the expression $49 - (z+4)^2$ as a product.

We hope that this article has been helpful in understanding how to write expressions as products. If you have any further questions or need additional help, please don't hesitate to ask.