Simplify The Expression:${ \frac{a^2 - B^2 + A - B}{a^2 + 2ab + B^2 - 1} }$

Mar 12, 2025 by ADMIN 77 views

Simplify the Expression: A Step-by-Step Guide

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Introduction

In this article, we will simplify the given expression using algebraic techniques. The expression is $\frac{a^2 - b^2 + a - b}{a^2 + 2ab + b^2 - 1}$ . We will break down the expression into smaller parts, simplify each part, and then combine them to get the final result.

Factorization of the Numerator

The numerator of the expression is $a^2 - b^2 + a - b$ . We can factorize this expression using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$ . Therefore, the numerator can be written as $(a + b)(a - b) + a - b$ .

Simplifying the Numerator

We can simplify the numerator further by combining the two terms: $(a + b)(a - b) + a - b = (a + b)(a - b) + (a - b)$ . We can factor out the common term $(a - b)$ from both terms: $(a + b)(a - b) + (a - b) = (a - b)((a + b) + 1)$ .

Factorization of the Denominator

The denominator of the expression is $a^2 + 2ab + b^2 - 1$ . We can factorize this expression using the perfect square formula: $a^2 + 2ab + b^2 = (a + b)^2$ . Therefore, the denominator can be written as $(a + b)^2 - 1$ .

Simplifying the Denominator

We can simplify the denominator further by factoring out the common term $(a + b)^2$ : $(a + b)^2 - 1 = (a + b)^2 - 1^2$ . We can use the difference of squares formula to simplify this expression: $(a + b)^2 - 1^2 = (a + b + 1)(a + b - 1)$ .

Simplifying the Expression

Now that we have simplified the numerator and denominator, we can simplify the expression by canceling out any common factors. The numerator is $(a - b)((a + b) + 1)$ and the denominator is $(a + b + 1)(a + b - 1)$ . We can cancel out the common factor $(a + b + 1)$ from both the numerator and denominator.

Final Result

After canceling out the common factor $(a + b + 1)$ , the expression becomes $\frac{(a - b)}{(a + b - 1)}$ . This is the simplified form of the given expression.

Conclusion

In this article, we simplified the given expression using algebraic techniques. We factorized the numerator and denominator, simplified each part, and then combined them to get the final result. The simplified expression is $\frac{(a - b)}{(a + b - 1)}$ .

Frequently Asked Questions

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$ .

Q: What is the perfect square formula?

A: The perfect square formula is $a^2 + 2ab + b^2 = (a + b)^2$ .

Q: How do I simplify the expression?

A: To simplify the expression, you need to factorize the numerator and denominator, simplify each part, and then combine them to get the final result.

Step-by-Step Guide

Step 1: Factorize the Numerator

Factorize the numerator using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$ .
Simplify the numerator further by combining the two terms: $(a + b)(a - b) + a - b = (a - b)((a + b) + 1)$ .

Step 2: Factorize the Denominator

Factorize the denominator using the perfect square formula: $a^2 + 2ab + b^2 = (a + b)^2$ .
Simplify the denominator further by factoring out the common term $(a + b)^2$ : $(a + b)^2 - 1 = (a + b)^2 - 1^2$ .
Use the difference of squares formula to simplify this expression: $(a + b)^2 - 1^2 = (a + b + 1)(a + b - 1)$ .

Step 3: Simplify the Expression

Cancel out any common factors between the numerator and denominator.
The numerator is $(a - b)((a + b) + 1)$ and the denominator is $(a + b + 1)(a + b - 1)$ .
Cancel out the common factor $(a + b + 1)$ from both the numerator and denominator.

Step 4: Final Result

The simplified expression is $\frac{(a - b)}{(a + b - 1)}$ .

Example Use Cases

Example 1

Suppose we have the expression $\frac{x^2 - 4 + x - 2}{x^2 + 6x + 9 - 1}$ . We can simplify this expression using the steps outlined above.

Example 2

Suppose we have the expression $\frac{y^2 - 9 + y - 3}{y^2 + 8y + 16 - 1}$ . We can simplify this expression using the steps outlined above.

Conclusion

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Introduction

In our previous article, we simplified the given expression using algebraic techniques. We factorized the numerator and denominator, simplified each part, and then combined them to get the final result. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the difference of squares formula?

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

Q: What is the perfect square formula?

A: The perfect square formula is $a^2 + 2ab + b^2 = (a + b)^2$ . This formula is used to factorize the perfect square.

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow these steps:

Factorize the numerator and denominator.
Simplify each part.
Combine the simplified parts to get the final result.

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

A: Some common mistakes to avoid when simplifying expressions include:

Not factoring the numerator and denominator properly.
Not simplifying each part correctly.
Not combining the simplified parts correctly.

Q: How do I know if I have simplified the expression correctly?

A: To check if you have simplified the expression correctly, you can:

Plug in some values for the variables and check if the expression simplifies to the expected result.
Use a calculator to check if the expression simplifies to the expected result.
Check if the simplified expression is in the simplest form possible.

Q: Can I simplify expressions with variables?

A: Yes, you can simplify expressions with variables. The steps for simplifying expressions with variables are the same as for simplifying expressions with constants.

Q: Can I simplify expressions with fractions?

A: Yes, you can simplify expressions with fractions. The steps for simplifying expressions with fractions are the same as for simplifying expressions with constants.

Q: How do I simplify expressions with exponents?

A: To simplify expressions with exponents, you need to follow these steps:

Simplify the expression inside the exponent.
Simplify the exponent itself.
Combine the simplified parts to get the final result.

Q: Can I simplify expressions with radicals?

A: Yes, you can simplify expressions with radicals. The steps for simplifying expressions with radicals are the same as for simplifying expressions with constants.

Example Use Cases

Example 1

Suppose we have the expression $\frac{x^2 - 4 + x - 2}{x^2 + 6x + 9 - 1}$ . We can simplify this expression using the steps outlined above.

Example 2

Suppose we have the expression $\frac{y^2 - 9 + y - 3}{y^2 + 8y + 16 - 1}$ . We can simplify this expression using the steps outlined above.

Conclusion

In this article, we answered some frequently asked questions related to simplifying expressions. We provided step-by-step guide and example use cases to help readers understand the concept better. Simplifying expressions is an important skill in mathematics, and it can be used in a variety of real-world applications.

Frequently Asked Questions

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2 - b^2 = (a + b)(a - b)$ .

Q: What is the perfect square formula?

A: The perfect square formula is $a^2 + 2ab + b^2 = (a + b)^2$ .

Q: How do I simplify the expression?

A: To simplify the expression, you need to follow these steps:

Factorize the numerator and denominator.
Simplify each part.
Combine the simplified parts to get the final result.

Step-by-Step Guide

Step 1: Factorize the Numerator

Factorize the numerator using the difference of squares formula: $a^2 - b^2 = (a + b)(a - b)$ .
Simplify the numerator further by combining the two terms: $(a + b)(a - b) + a - b = (a - b)((a + b) + 1)$ .

Step 2: Factorize the Denominator

Factorize the denominator using the perfect square formula: $a^2 + 2ab + b^2 = (a + b)^2$ .
Simplify the denominator further by factoring out the common term $(a + b)^2$ : $(a + b)^2 - 1 = (a + b)^2 - 1^2$ .
Use the difference of squares formula to simplify this expression: $(a + b)^2 - 1^2 = (a + b + 1)(a + b - 1)$ .

Step 3: Simplify the Expression

Cancel out any common factors between the numerator and denominator.
The numerator is $(a - b)((a + b) + 1)$ and the denominator is $(a + b + 1)(a + b - 1)$ .
Cancel out the common factor $(a + b + 1)$ from both the numerator and denominator.

Step 4: Final Result

The simplified expression is $\frac{(a - b)}{(a + b - 1)}$ .

Example Use Cases

Example 1

Suppose we have the expression $\frac{x^2 - 4 + x - 2}{x^2 + 6x + 9 - 1}$ . We can simplify this expression using the steps outlined above.

Example 2

Suppose we have the expression $\frac{y^2 - 9 + y - 3}{y^2 + 8y + 16 - 1}$ . We can simplify this expression using the steps outlined above.

Conclusion

In this article, we provided a step-by-step guide and example use cases to help readers understand the concept of simplifying expressions. We also answered some frequently asked questions related to simplifying expressions. Simplifying expressions is an important skill in mathematics, and it can be used in a variety of real-world applications.