(4/3m-3/4m)^2+2mn=16/9m2+9/16n2

Jasonbradley

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Jun 16, 2024

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Simplifying and Solving Algebraic Expressions

This article explores how to simplify and solve the given algebraic expression: (4/3m - 3/4m)^2 + 2mn = 16/9m^2 + 9/16n^2.

Simplifying the Expression

1. Combine like terms: Begin by simplifying the expression inside the parentheses.

   (4/3m - 3/4m)^2 + 2mn = (16/12m - 9/12m)^2 + 2mn
   

   This simplifies to:

   (7/12m)^2 + 2mn 
   

2. Square the term: Next, square the term (7/12m).

   (7/12m)^2 + 2mn = 49/144m^2 + 2mn
   

3. Compare the expressions: Now compare the simplified expression with the given expression on the right side of the equation:

   49/144m^2 + 2mn = 16/9m^2 + 9/16n^2
   

Solving the Equation

To solve this equation, we need to isolate the variables m and n.

1. Rearrange the terms: Move all terms to one side of the equation.

   49/144m^2 + 2mn - 16/9m^2 - 9/16n^2 = 0
   

2. Simplify: Combine like terms.

   -119/144m^2 + 2mn - 9/16n^2 = 0
   

3. Factor out common factors: Notice that we can factor out a -1/144 from the left side.

   -1/144(-119m^2 + 288mn - 126n^2) = 0
   

   This gives us:

   119m^2 - 288mn + 126n^2 = 0 
   

4. Factor the quadratic: The equation is now a quadratic in terms of m and n. Factoring this equation would require advanced techniques and is beyond the scope of this article.

Conclusion

The simplified form of the expression is 119m^2 - 288mn + 126n^2 = 0. Solving this equation to find the values of m and n requires further factoring and algebraic manipulation. This article provides a step-by-step guide to simplifying the given expression, highlighting the process of combining like terms and squaring the terms.