Solve (x+1/3y)(x-1/3y)-(-x-y)(x-y)

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

Evaluate

$$-\frac{10y^{2}}{9}+2x^{2}$$

Solution Steps

Consider \left(x+\frac{1}{3}y\right)\left(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}.

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

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

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

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

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

Apply the distributive property by multiplying each term of -x-y by each term of x-y.

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

Multiply -1 and -1 to get 1.

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

Combine xy and -yx to get 0.

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

To find the opposite of \left(-x\right)x+y^{2}, find the opposite of each term.

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

Multiply -1 and -1 to get 1.

$$x^{2}-\frac{1}{9}y^{2}+xx-y^{2}$$

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

$$x^{2}-\frac{1}{9}y^{2}+x^{2}-y^{2}$$

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

$$2x^{2}-\frac{1}{9}y^{2}-y^{2}$$

Combine -\frac{1}{9}y^{2} and -y^{2} to get -\frac{10}{9}y^{2}.

$$2x^{2}-\frac{10}{9}y^{2}$$

Expand

$$-\frac{10y^{2}}{9}+2x^{2}$$

Solution Steps

Consider \left(x+\frac{1}{3}y\right)\left(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}.

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

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

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

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

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

Apply the distributive property by multiplying each term of -x-y by each term of x-y.

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

Multiply -1 and -1 to get 1.

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

Combine xy and -yx to get 0.

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

To find the opposite of \left(-x\right)x+y^{2}, find the opposite of each term.

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

Multiply -1 and -1 to get 1.

$$x^{2}-\frac{1}{9}y^{2}+xx-y^{2}$$

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

$$x^{2}-\frac{1}{9}y^{2}+x^{2}-y^{2}$$

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

$$2x^{2}-\frac{1}{9}y^{2}-y^{2}$$

Combine -\frac{1}{9}y^{2} and -y^{2} to get -\frac{10}{9}y^{2}.

$$2x^{2}-\frac{10}{9}y^{2}$$