Math Problem Statement

Perform the indicated operations for the expression: 2ab/(a^2 - b^2) - b/(a - b) + 6/2

Solution

We are asked to perform the indicated operations for the given expression:

2aba2b2bab+62\frac{2ab}{a^2 - b^2} - \frac{b}{a - b} + \frac{6}{2}

Step-by-Step Solution:

  1. Simplify the terms:

    • The first term 2aba2b2\frac{2ab}{a^2 - b^2}: Notice that a2b2a^2 - b^2 is a difference of squares and can be factored: a2b2=(ab)(a+b)a^2 - b^2 = (a - b)(a + b) So the first term becomes: 2ab(ab)(a+b)\frac{2ab}{(a - b)(a + b)}

    • The second term bab\frac{b}{a - b} remains unchanged.

    • The third term 62\frac{6}{2} simplifies to 33.

  2. Common Denominator:

    To combine the first two fractions, find a common denominator. The first term has a denominator of (ab)(a+b)(a - b)(a + b), while the second term has a denominator of aba - b. The least common denominator (LCD) is (ab)(a+b)(a - b)(a + b).

    Rewrite the second term to have this common denominator: bab=b(a+b)(ab)(a+b)\frac{b}{a - b} = \frac{b(a + b)}{(a - b)(a + b)}

  3. Combine the first two terms:

    Now, subtract the two fractions: 2ab(ab)(a+b)b(a+b)(ab)(a+b)=2abb(a+b)(ab)(a+b)\frac{2ab}{(a - b)(a + b)} - \frac{b(a + b)}{(a - b)(a + b)} = \frac{2ab - b(a + b)}{(a - b)(a + b)} Expand the numerator of the second term: 2abb(a+b)=2ab(ba+b2)=2abbab2=abb22ab - b(a + b) = 2ab - (ba + b^2) = 2ab - ba - b^2 = ab - b^2 So, the combined fraction is: abb2(ab)(a+b)\frac{ab - b^2}{(a - b)(a + b)}

  4. Add the constant term:

    Now add the constant 33 to the fraction: abb2(ab)(a+b)+3\frac{ab - b^2}{(a - b)(a + b)} + 3

    We can leave this as the final answer unless a further simplification is requested.

Final Answer:

abb2(ab)(a+b)+3\frac{ab - b^2}{(a - b)(a + b)} + 3

Let me know if you'd like further details or clarifications!


Related Questions:

  1. How can we simplify rational expressions more effectively?
  2. What are common factoring techniques, like the difference of squares, and when do they apply?
  3. How do we find a common denominator for more complex fractions?
  4. How would this problem change if we were given numerical values for aa and bb?
  5. How can we manipulate algebraic fractions involving polynomials?

Tip:

When dealing with expressions that include a difference of squares, remember that a2b2a^2 - b^2 always factors into (ab)(a+b)(a - b)(a + b), which can simplify many algebraic operations.

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Math Problem Analysis

Mathematical Concepts

Algebra
Simplifying Rational Expressions
Factoring

Formulas

Difference of Squares: a^2 - b^2 = (a - b)(a + b)

Theorems

Factoring Theorem

Suitable Grade Level

Grades 10-12