Solve (x-y)^2-12(y^2-x^2)+4(x+y)^2

$$( x - y ) ^ { 2 } - 12 ( y ^ { 2 } - x ^ { 2 } ) + 4 ( x + y ) ^ { 2 }$$

Evaluate

$$17x^{2}+6xy-7y^{2}$$

Solution Steps

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

$$x^{2}-2xy+y^{2}-12\left(y^{2}-x^{2}\right)+4\left(x+y\right)^{2}$$

Use the distributive property to multiply -12 by y^{2}-x^{2}.

$$x^{2}-2xy+y^{2}-12y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine y^{2} and -12y^{2} to get -11y^{2}.

$$x^{2}-2xy-11y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine x^{2} and 12x^{2} to get 13x^{2}.

$$13x^{2}-2xy-11y^{2}+4\left(x+y\right)^{2}$$

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

$$13x^{2}-2xy-11y^{2}+4\left(x^{2}+2xy+y^{2}\right)$$

Use the distributive property to multiply 4 by x^{2}+2xy+y^{2}.

$$13x^{2}-2xy-11y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 13x^{2} and 4x^{2} to get 17x^{2}.

$$17x^{2}-2xy-11y^{2}+8xy+4y^{2}$$

Combine -2xy and 8xy to get 6xy.

$$17x^{2}+6xy-11y^{2}+4y^{2}$$

Combine -11y^{2} and 4y^{2} to get -7y^{2}.

$$17x^{2}+6xy-7y^{2}$$

Expand

$$17x^{2}+6xy-7y^{2}$$

Solution Steps

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

$$x^{2}-2xy+y^{2}-12\left(y^{2}-x^{2}\right)+4\left(x+y\right)^{2}$$

Use the distributive property to multiply -12 by y^{2}-x^{2}.

$$x^{2}-2xy+y^{2}-12y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine y^{2} and -12y^{2} to get -11y^{2}.

$$x^{2}-2xy-11y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine x^{2} and 12x^{2} to get 13x^{2}.

$$13x^{2}-2xy-11y^{2}+4\left(x+y\right)^{2}$$

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

$$13x^{2}-2xy-11y^{2}+4\left(x^{2}+2xy+y^{2}\right)$$

Use the distributive property to multiply 4 by x^{2}+2xy+y^{2}.

$$13x^{2}-2xy-11y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 13x^{2} and 4x^{2} to get 17x^{2}.

$$17x^{2}-2xy-11y^{2}+8xy+4y^{2}$$

Combine -2xy and 8xy to get 6xy.

$$17x^{2}+6xy-11y^{2}+4y^{2}$$

Combine -11y^{2} and 4y^{2} to get -7y^{2}.

$$17x^{2}+6xy-7y^{2}$$