Math Problem Statement

Subtract the rational expressions and simplify the result: (10 / (25a^2 - b^2)) - (1 / (5a^2 - ab))

Solution

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The expression seems to be:

1025a2b215a2ab\frac{10}{25a^2 - b^2} - \frac{1}{5a^2 - ab}

The next step is to find a common denominator and simplify the result.

Step-by-step Solution:

  1. Factor each denominator:

    • 25a2b225a^2 - b^2 is a difference of squares, which factors as: 25a2b2=(5ab)(5a+b)25a^2 - b^2 = (5a - b)(5a + b)
    • 5a2ab5a^2 - ab can be factored by taking out the common factor of aa: 5a2ab=a(5ab)5a^2 - ab = a(5a - b)
  2. Find a common denominator: The least common denominator (LCD) between (5ab)(5a+b)(5a - b)(5a + b) and a(5ab)a(5a - b) is: a(5ab)(5a+b)a(5a - b)(5a + b)

  3. Rewrite each fraction with the LCD: 10(5ab)(5a+b)=10aa(5ab)(5a+b)\frac{10}{(5a - b)(5a + b)} = \frac{10a}{a(5a - b)(5a + b)} 1a(5ab)=(5a+b)a(5ab)(5a+b)\frac{1}{a(5a - b)} = \frac{(5a + b)}{a(5a - b)(5a + b)}

  4. Subtract the numerators: Now that the denominators are the same, we subtract the numerators: 10a(5a+b)a(5ab)(5a+b)\frac{10a - (5a + b)}{a(5a - b)(5a + b)} Simplifying the numerator: 10a5ab=5ab10a - 5a - b = 5a - b

  5. Final result: The final expression is: 5aba(5ab)(5a+b)\frac{5a - b}{a(5a - b)(5a + b)} The 5ab5a - b in the numerator and denominator cancel out (as long as 5ab5a \neq b), leaving: 1a(5a+b)\frac{1}{a(5a + b)}

This is the simplified form of the given expression.


Do you want any additional details or clarifications on this?

Here are five related questions for further practice:

  1. How do you simplify rational expressions with multiple terms?
  2. What is the difference of squares formula?
  3. How do you find a least common denominator (LCD)?
  4. How do you handle subtraction when simplifying rational expressions?
  5. What steps are involved in factoring quadratic expressions?

Tip: When dealing with rational expressions, always check for common factors that can be canceled out after you find a common denominator!

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Math Problem Analysis

Mathematical Concepts

Algebra
Rational Expressions
Factoring
Common Denominator

Formulas

Difference of squares: a^2 - b^2 = (a - b)(a + b)
Factoring quadratic expressions

Theorems

Properties of Rational Expressions
Factoring Theorem

Suitable Grade Level

Grades 9-11