Solve (x-y)^2-8(x^2-y^2)+16(x+y)^2

$$( x - y ) ^ { 2 } - 8 ( x ^ { 2 } - y ^ { 2 } ) + 16 ( x + y ) ^ { 2 }$$

Evaluate

$$\left(3x+5y\right)^{2}$$

Solution Steps

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

$$x^{2}-2xy+y^{2}-8\left(x^{2}-y^{2}\right)+16\left(x+y\right)^{2}$$

Use the distributive property to multiply -8 by x^{2}-y^{2}.

$$x^{2}-2xy+y^{2}-8x^{2}+8y^{2}+16\left(x+y\right)^{2}$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}-2xy+y^{2}+8y^{2}+16\left(x+y\right)^{2}$$

Combine y^{2} and 8y^{2} to get 9y^{2}.

$$-7x^{2}-2xy+9y^{2}+16\left(x+y\right)^{2}$$

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

$$-7x^{2}-2xy+9y^{2}+16\left(x^{2}+2xy+y^{2}\right)$$

Use the distributive property to multiply 16 by x^{2}+2xy+y^{2}.

$$-7x^{2}-2xy+9y^{2}+16x^{2}+32xy+16y^{2}$$

Combine -7x^{2} and 16x^{2} to get 9x^{2}.

$$9x^{2}-2xy+9y^{2}+32xy+16y^{2}$$

Combine -2xy and 32xy to get 30xy.

$$9x^{2}+30xy+9y^{2}+16y^{2}$$

Combine 9y^{2} and 16y^{2} to get 25y^{2}.

$$9x^{2}+30xy+25y^{2}$$

Expand

$$9x^{2}+30xy+25y^{2}$$

Solution Steps

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

$$x^{2}-2xy+y^{2}-8\left(x^{2}-y^{2}\right)+16\left(x+y\right)^{2}$$

Use the distributive property to multiply -8 by x^{2}-y^{2}.

$$x^{2}-2xy+y^{2}-8x^{2}+8y^{2}+16\left(x+y\right)^{2}$$

Combine x^{2} and -8x^{2} to get -7x^{2}.

$$-7x^{2}-2xy+y^{2}+8y^{2}+16\left(x+y\right)^{2}$$

Combine y^{2} and 8y^{2} to get 9y^{2}.

$$-7x^{2}-2xy+9y^{2}+16\left(x+y\right)^{2}$$

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

$$-7x^{2}-2xy+9y^{2}+16\left(x^{2}+2xy+y^{2}\right)$$

Use the distributive property to multiply 16 by x^{2}+2xy+y^{2}.

$$-7x^{2}-2xy+9y^{2}+16x^{2}+32xy+16y^{2}$$

Combine -7x^{2} and 16x^{2} to get 9x^{2}.

$$9x^{2}-2xy+9y^{2}+32xy+16y^{2}$$

Combine -2xy and 32xy to get 30xy.

$$9x^{2}+30xy+9y^{2}+16y^{2}$$

Combine 9y^{2} and 16y^{2} to get 25y^{2}.

$$9x^{2}+30xy+25y^{2}$$