Resolver 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)

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

Calcular

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

Pasos da solución

Usar teorema binomial \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para expandir \left(\frac{3}{4}x+\frac{1}{3}y\right)^{2}.

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

Combina -\frac{1}{3}y^{2} e \frac{1}{9}y^{2} para obter -\frac{2}{9}y^{2}.

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

Usa a propiedade distributiva para multiplicar 2 por \frac{3}{4}x+\frac{1}{3}y.

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

Usa a propiedade distributiva para multiplicar \frac{3}{2}x+\frac{2}{3}y por \frac{3}{4}x-\frac{1}{3}y e combina os termos semellantes.

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

Combina \frac{9}{16}x^{2} e \frac{9}{8}x^{2} para obter \frac{27}{16}x^{2}.

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

Combina -\frac{2}{9}y^{2} e -\frac{2}{9}y^{2} para obter -\frac{4}{9}y^{2}.

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

Expandir

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

Pasos da solución

Usar teorema binomial \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para expandir \left(\frac{3}{4}x+\frac{1}{3}y\right)^{2}.

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

Combina -\frac{1}{3}y^{2} e \frac{1}{9}y^{2} para obter -\frac{2}{9}y^{2}.

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

Usa a propiedade distributiva para multiplicar 2 por \frac{3}{4}x+\frac{1}{3}y.

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

Usa a propiedade distributiva para multiplicar \frac{3}{2}x+\frac{2}{3}y por \frac{3}{4}x-\frac{1}{3}y e combina os termos semellantes.

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

Combina \frac{9}{16}x^{2} e \frac{9}{8}x^{2} para obter \frac{27}{16}x^{2}.

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

Combina -\frac{2}{9}y^{2} e -\frac{2}{9}y^{2} para obter -\frac{4}{9}y^{2}.

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