Solve left(1/2x+y-1right)left(-1/2x-y+1right)+{left(1/2x-y-1right)}^2-4y

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

Evaluate

$$-2xy$$

Solution Steps

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

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

Square \frac{1}{2}x-y-1.

$$-\frac{1}{4}x^{2}-xy+x-y^{2}+2y-1+\frac{1}{4}x^{2}-xy-x+y^{2}+2y+1-4y$$

Combine -\frac{1}{4}x^{2} and \frac{1}{4}x^{2} to get 0.

$$-xy+x-y^{2}+2y-1-xy-x+y^{2}+2y+1-4y$$

Combine -xy and -xy to get -2xy.

$$-2xy+x-y^{2}+2y-1-x+y^{2}+2y+1-4y$$

Combine x and -x to get 0.

$$-2xy-y^{2}+2y-1+y^{2}+2y+1-4y$$

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

$$-2xy+2y-1+2y+1-4y$$

Combine 2y and 2y to get 4y.

$$-2xy+4y-1+1-4y$$

Add -1 and 1 to get 0.

$$-2xy+4y-4y$$

Combine 4y and -4y to get 0.

$$-2xy$$

Expand

$$-2xy$$

Solution Steps

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

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

Square \frac{1}{2}x-y-1.

$$-\frac{1}{4}x^{2}-xy+x-y^{2}+2y-1+\frac{1}{4}x^{2}-xy-x+y^{2}+2y+1-4y$$

Combine -\frac{1}{4}x^{2} and \frac{1}{4}x^{2} to get 0.

$$-xy+x-y^{2}+2y-1-xy-x+y^{2}+2y+1-4y$$

Combine -xy and -xy to get -2xy.

$$-2xy+x-y^{2}+2y-1-x+y^{2}+2y+1-4y$$

Combine x and -x to get 0.

$$-2xy-y^{2}+2y-1+y^{2}+2y+1-4y$$

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

$$-2xy+2y-1+2y+1-4y$$

Combine 2y and 2y to get 4y.

$$-2xy+4y-1+1-4y$$

Add -1 and 1 to get 0.

$$-2xy+4y-4y$$

Combine 4y and -4y to get 0.

$$-2xy$$