Solve (3/4x+2/5y)(-2/5y+3/4x)

$$( \frac { 3 } { 4 } x + \frac { 2 } { 5 } y ) ( - \frac { 2 } { 5 } y + \frac { 3 } { 4 } x )$$

Evaluate

$$\frac{9x^{2}}{16}-\frac{4y^{2}}{25}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{3}{4}x+\frac{2}{5}y by each term of -\frac{2}{5}y+\frac{3}{4}x.

$$\frac{3}{4}x\left(-\frac{2}{5}\right)y+\frac{3}{4}x\times \frac{3}{4}x+\frac{2}{5}y\left(-\frac{2}{5}\right)y+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply x and x to get x^{2}.

$$\frac{3}{4}x\left(-\frac{2}{5}\right)y+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y\left(-\frac{2}{5}\right)y+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply y and y to get y^{2}.

$$\frac{3}{4}x\left(-\frac{2}{5}\right)y+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{3}{4} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(-2\right)}{4\times 5}xy+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Do the multiplications in the fraction \frac{3\left(-2\right)}{4\times 5}.

$$\frac{-6}{20}xy+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Reduce the fraction \frac{-6}{20} to lowest terms by extracting and canceling out 2.

$$-\frac{3}{10}xy+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{3}{4} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{10}xy+\frac{3\times 3}{4\times 4}x^{2}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Do the multiplications in the fraction \frac{3\times 3}{4\times 4}.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{2}{5} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}+\frac{2\left(-2\right)}{5\times 5}y^{2}+\frac{2}{5}y\times \frac{3}{4}x$$

Do the multiplications in the fraction \frac{2\left(-2\right)}{5\times 5}.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}+\frac{-4}{25}y^{2}+\frac{2}{5}y\times \frac{3}{4}x$$

Fraction \frac{-4}{25} can be rewritten as -\frac{4}{25} by extracting the negative sign.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{2}{5} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{2\times 3}{5\times 4}yx$$

Do the multiplications in the fraction \frac{2\times 3}{5\times 4}.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{6}{20}yx$$

Reduce the fraction \frac{6}{20} to lowest terms by extracting and canceling out 2.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{3}{10}yx$$

Combine -\frac{3}{10}xy and \frac{3}{10}yx to get 0.

$$\frac{9}{16}x^{2}-\frac{4}{25}y^{2}$$

Expand

$$\frac{9x^{2}}{16}-\frac{4y^{2}}{25}$$

Solution Steps

Apply the distributive property by multiplying each term of \frac{3}{4}x+\frac{2}{5}y by each term of -\frac{2}{5}y+\frac{3}{4}x.

$$\frac{3}{4}x\left(-\frac{2}{5}\right)y+\frac{3}{4}x\times \frac{3}{4}x+\frac{2}{5}y\left(-\frac{2}{5}\right)y+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply x and x to get x^{2}.

$$\frac{3}{4}x\left(-\frac{2}{5}\right)y+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y\left(-\frac{2}{5}\right)y+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply y and y to get y^{2}.

$$\frac{3}{4}x\left(-\frac{2}{5}\right)y+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{3}{4} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3\left(-2\right)}{4\times 5}xy+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Do the multiplications in the fraction \frac{3\left(-2\right)}{4\times 5}.

$$\frac{-6}{20}xy+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Reduce the fraction \frac{-6}{20} to lowest terms by extracting and canceling out 2.

$$-\frac{3}{10}xy+\frac{3}{4}x^{2}\times \frac{3}{4}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{3}{4} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{10}xy+\frac{3\times 3}{4\times 4}x^{2}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Do the multiplications in the fraction \frac{3\times 3}{4\times 4}.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}+\frac{2}{5}y^{2}\left(-\frac{2}{5}\right)+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{2}{5} times -\frac{2}{5} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}+\frac{2\left(-2\right)}{5\times 5}y^{2}+\frac{2}{5}y\times \frac{3}{4}x$$

Do the multiplications in the fraction \frac{2\left(-2\right)}{5\times 5}.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}+\frac{-4}{25}y^{2}+\frac{2}{5}y\times \frac{3}{4}x$$

Fraction \frac{-4}{25} can be rewritten as -\frac{4}{25} by extracting the negative sign.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{2}{5}y\times \frac{3}{4}x$$

Multiply \frac{2}{5} times \frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{2\times 3}{5\times 4}yx$$

Do the multiplications in the fraction \frac{2\times 3}{5\times 4}.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{6}{20}yx$$

Reduce the fraction \frac{6}{20} to lowest terms by extracting and canceling out 2.

$$-\frac{3}{10}xy+\frac{9}{16}x^{2}-\frac{4}{25}y^{2}+\frac{3}{10}yx$$

Combine -\frac{3}{10}xy and \frac{3}{10}yx to get 0.

$$\frac{9}{16}x^{2}-\frac{4}{25}y^{2}$$