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Simplify the following expression: (36x2 - 9y2) / 4xy - 20xy / (6x - 3y)

Mathematics

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Answer

27(2x−y)2(2x+y)−80x2y212xy(2x−y)\frac{27(2x-y)^2(2x+y) - 80x^2y^2}{12xy(2x-y)}

Step-by-step solution

5 steps

Here's the solution to the problem.

36x2−9y24xy−20xy6x−3y\frac{36x^2 - 9y^2}{4xy} - \frac{20xy}{6x - 3y}

  1. Factor the expressions in the numerators and denominators.

    The numerator of the first fraction is a difference of squares after factoring out 9:

    36x2−9y2=9(4x2−y2)=9((2x)2−y2)=9(2x−y)(2x+y)36x^2 - 9y^2 = 9(4x^2 - y^2) = 9((2x)^2 - y^2) = 9(2x-y)(2x+y)

    The denominator of the second fraction can be factored:

    6x−3y=3(2x−y)6x - 3y = 3(2x - y)

  2. Rewrite the expression with the factored terms.

    9(2x−y)(2x+y)4xy−20xy3(2x−y)\frac{9(2x-y)(2x+y)}{4xy} - \frac{20xy}{3(2x - y)}

  3. Determine the least common denominator (LCD) for the two fractions.

    The denominators are 4xy4xy and 3(2x−y)3(2x-y). The LCD is 12xy(2x−y)12xy(2x-y).

  4. Rewrite each fraction with the LCD.

    For the first fraction, multiply the numerator and denominator by 3(2x−y)3(2x-y):

    9(2x−y)(2x+y)4xy×3(2x−y)3(2x−y)=27(2x−y)2(2x+y)12xy(2x−y)\frac{9(2x-y)(2x+y)}{4xy} \times \frac{3(2x-y)}{3(2x-y)} = \frac{27(2x-y)^2(2x+y)}{12xy(2x-y)}

    For the second fraction, multiply the numerator and denominator by 4xy4xy:

    20xy3(2x−y)×4xy4xy=80x2y212xy(2x−y)\frac{20xy}{3(2x - y)} \times \frac{4xy}{4xy} = \frac{80x^2y^2}{12xy(2x-y)}

  5. Subtract the fractions.

    27(2x−y)2(2x+y)12xy(2x−y)−80x2y212xy(2x−y)=27(2x−y)2(2x+y)−80x2y212xy(2x−y)\frac{27(2x-y)^2(2x+y)}{12xy(2x-y)} - \frac{80x^2y^2}{12xy(2x-y)} = \frac{27(2x-y)^2(2x+y) - 80x^2y^2}{12xy(2x-y)}

    The simplified expression is 27(2x−y)2(2x+y)−80x2y212xy(2x−y)\frac{27(2x-y)^2(2x+y) - 80x^2y^2}{12xy(2x-y)}.

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