Solve 3/4x-1/3y^2+(3/4x+1/3y)^2+2(frac{3}{4}x+frac{1}{3}y)(frac{3}{4}x-frac{1}{3}y)

$$\frac { 3 } { 4 } x - \frac { 1 } { 3 } y ^ { 2 } + ( \frac { 3 } { 4 } x + \frac { 1 } { 3 } y ) ^ { 2 } + 2 ( \frac { 3 } { 4 } x + \frac { 1 } { 3 } y ) ( \frac { 3 } { 4 } x - \frac { 1 } { 3 } y )$$

Evaluate

$$\frac{xy}{2}+\frac{27x^{2}}{16}-\frac{4y^{2}}{9}+\frac{3x}{4}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{3}{4}x+\frac{1}{3}y\right)^{2}.

$$\frac{3}{4}x-\frac{1}{3}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+\frac{1}{9}y^{2}+2\left(\frac{3}{4}x+\frac{1}{3}y\right)\left(\frac{3}{4}x-\frac{1}{3}y\right)$$

Combine -\frac{1}{3}y^{2} and \frac{1}{9}y^{2} to get -\frac{2}{9}y^{2}.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+2\left(\frac{3}{4}x+\frac{1}{3}y\right)\left(\frac{3}{4}x-\frac{1}{3}y\right)$$

Use the distributive property to multiply 2 by \frac{3}{4}x+\frac{1}{3}y.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+\left(\frac{3}{2}x+\frac{2}{3}y\right)\left(\frac{3}{4}x-\frac{1}{3}y\right)$$

Use the distributive property to multiply \frac{3}{2}x+\frac{2}{3}y by \frac{3}{4}x-\frac{1}{3}y and combine like terms.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+\frac{9}{8}x^{2}-\frac{2}{9}y^{2}$$

Combine \frac{9}{16}x^{2} and \frac{9}{8}x^{2} to get \frac{27}{16}x^{2}.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{27}{16}x^{2}+\frac{1}{2}xy-\frac{2}{9}y^{2}$$

Combine -\frac{2}{9}y^{2} and -\frac{2}{9}y^{2} to get -\frac{4}{9}y^{2}.

$$\frac{3}{4}x-\frac{4}{9}y^{2}+\frac{27}{16}x^{2}+\frac{1}{2}xy$$

Expand

$$\frac{xy}{2}+\frac{27x^{2}}{16}-\frac{4y^{2}}{9}+\frac{3x}{4}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(\frac{3}{4}x+\frac{1}{3}y\right)^{2}.

$$\frac{3}{4}x-\frac{1}{3}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+\frac{1}{9}y^{2}+2\left(\frac{3}{4}x+\frac{1}{3}y\right)\left(\frac{3}{4}x-\frac{1}{3}y\right)$$

Combine -\frac{1}{3}y^{2} and \frac{1}{9}y^{2} to get -\frac{2}{9}y^{2}.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+2\left(\frac{3}{4}x+\frac{1}{3}y\right)\left(\frac{3}{4}x-\frac{1}{3}y\right)$$

Use the distributive property to multiply 2 by \frac{3}{4}x+\frac{1}{3}y.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+\left(\frac{3}{2}x+\frac{2}{3}y\right)\left(\frac{3}{4}x-\frac{1}{3}y\right)$$

Use the distributive property to multiply \frac{3}{2}x+\frac{2}{3}y by \frac{3}{4}x-\frac{1}{3}y and combine like terms.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{9}{16}x^{2}+\frac{1}{2}xy+\frac{9}{8}x^{2}-\frac{2}{9}y^{2}$$

Combine \frac{9}{16}x^{2} and \frac{9}{8}x^{2} to get \frac{27}{16}x^{2}.

$$\frac{3}{4}x-\frac{2}{9}y^{2}+\frac{27}{16}x^{2}+\frac{1}{2}xy-\frac{2}{9}y^{2}$$

Combine -\frac{2}{9}y^{2} and -\frac{2}{9}y^{2} to get -\frac{4}{9}y^{2}.

$$\frac{3}{4}x-\frac{4}{9}y^{2}+\frac{27}{16}x^{2}+\frac{1}{2}xy$$