Solve left(1/2x-1/3yright)left(1/2x+1/3yright)-frac{8}{9}y^2+frac{3}{4}x^2+left(2x-yright)left(-2x-yright)

$$\left( \frac{ 1 }{ 2 } x- \frac{ 1 }{ 3 } y \right) \left( \frac{ 1 }{ 2 } x+ \frac{ 1 }{ 3 } y \right) - \frac{ 8 }{ 9 } { y }^{ 2 } + \frac{ 3 }{ 4 } { x }^{ 2 } + \left( 2x-y \right) \left( -2x-y \right)$$

Evaluate

$$-3x^{2}$$

Solution Steps

Consider \left(\frac{1}{2}x-\frac{1}{3}y\right)\left(\frac{1}{2}x+\frac{1}{3}y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\frac{1}{2}x\right)^{2}-\left(\frac{1}{3}y\right)^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Expand \left(\frac{1}{2}x\right)^{2}.

$$\left(\frac{1}{2}\right)^{2}x^{2}-\left(\frac{1}{3}y\right)^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{4}x^{2}-\left(\frac{1}{3}y\right)^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Expand \left(\frac{1}{3}y\right)^{2}.

$$\frac{1}{4}x^{2}-\left(\frac{1}{3}\right)^{2}y^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{1}{4}x^{2}-\frac{1}{9}y^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Combine -\frac{1}{9}y^{2} and -\frac{8}{9}y^{2} to get -y^{2}.

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

Combine \frac{1}{4}x^{2} and \frac{3}{4}x^{2} to get x^{2}.

$$x^{2}-y^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Use the distributive property to multiply 2x-y by -2x-y and combine like terms.

$$x^{2}-y^{2}-4x^{2}+y^{2}$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-y^{2}+y^{2}$$

Combine -y^{2} and y^{2} to get 0.

$$-3x^{2}$$

Expand

$$-3x^{2}$$

Solution Steps

Consider \left(\frac{1}{2}x-\frac{1}{3}y\right)\left(\frac{1}{2}x+\frac{1}{3}y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\left(\frac{1}{2}x\right)^{2}-\left(\frac{1}{3}y\right)^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Expand \left(\frac{1}{2}x\right)^{2}.

$$\left(\frac{1}{2}\right)^{2}x^{2}-\left(\frac{1}{3}y\right)^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Calculate \frac{1}{2} to the power of 2 and get \frac{1}{4}.

$$\frac{1}{4}x^{2}-\left(\frac{1}{3}y\right)^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Expand \left(\frac{1}{3}y\right)^{2}.

$$\frac{1}{4}x^{2}-\left(\frac{1}{3}\right)^{2}y^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Calculate \frac{1}{3} to the power of 2 and get \frac{1}{9}.

$$\frac{1}{4}x^{2}-\frac{1}{9}y^{2}-\frac{8}{9}y^{2}+\frac{3}{4}x^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Combine -\frac{1}{9}y^{2} and -\frac{8}{9}y^{2} to get -y^{2}.

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

Combine \frac{1}{4}x^{2} and \frac{3}{4}x^{2} to get x^{2}.

$$x^{2}-y^{2}+\left(2x-y\right)\left(-2x-y\right)$$

Use the distributive property to multiply 2x-y by -2x-y and combine like terms.

$$x^{2}-y^{2}-4x^{2}+y^{2}$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-3x^{2}-y^{2}+y^{2}$$

Combine -y^{2} and y^{2} to get 0.

$$-3x^{2}$$