Solve 13left(x-2yright)^2-8left(x-2yright)left(2x+yright)+7left(2x+yright)^2

$$13 { \left(x-2y \right) }^{ 2 } -8 \left( x-2y \right) \left( 2x+y \right) +7 { \left(2x+y \right) }^{ 2 }$$

Evaluate

$$25\left(x^{2}+3y^{2}\right)$$

Solution Steps

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

$$13\left(x^{2}-4xy+4y^{2}\right)-8\left(x-2y\right)\left(2x+y\right)+7\left(2x+y\right)^{2}$$

Use the distributive property to multiply 13 by x^{2}-4xy+4y^{2}.

$$13x^{2}-52xy+52y^{2}-8\left(x-2y\right)\left(2x+y\right)+7\left(2x+y\right)^{2}$$

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

$$13x^{2}-52xy+52y^{2}-8\left(x-2y\right)\left(2x+y\right)+7\left(4x^{2}+4xy+y^{2}\right)$$

Use the distributive property to multiply 7 by 4x^{2}+4xy+y^{2}.

$$13x^{2}-52xy+52y^{2}-8\left(x-2y\right)\left(2x+y\right)+28x^{2}+28xy+7y^{2}$$

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

$$13x^{2}-52xy+52y^{2}+\left(-8x+16y\right)\left(2x+y\right)+28x^{2}+28xy+7y^{2}$$

Use the distributive property to multiply -8x+16y by 2x+y and combine like terms.

$$13x^{2}-52xy+52y^{2}-16x^{2}+24xy+16y^{2}+28x^{2}+28xy+7y^{2}$$

Combine 13x^{2} and -16x^{2} to get -3x^{2}.

$$-3x^{2}-52xy+52y^{2}+24xy+16y^{2}+28x^{2}+28xy+7y^{2}$$

Combine -52xy and 24xy to get -28xy.

$$-3x^{2}-28xy+52y^{2}+16y^{2}+28x^{2}+28xy+7y^{2}$$

Combine 52y^{2} and 16y^{2} to get 68y^{2}.

$$-3x^{2}-28xy+68y^{2}+28x^{2}+28xy+7y^{2}$$

Combine -3x^{2} and 28x^{2} to get 25x^{2}.

$$25x^{2}-28xy+68y^{2}+28xy+7y^{2}$$

Combine -28xy and 28xy to get 0.

$$25x^{2}+68y^{2}+7y^{2}$$

Combine 68y^{2} and 7y^{2} to get 75y^{2}.

$$25x^{2}+75y^{2}$$

Expand

$$25x^{2}+75y^{2}$$

Solution Steps

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

$$13\left(x^{2}-4xy+4y^{2}\right)-8\left(x-2y\right)\left(2x+y\right)+7\left(2x+y\right)^{2}$$

Use the distributive property to multiply 13 by x^{2}-4xy+4y^{2}.

$$13x^{2}-52xy+52y^{2}-8\left(x-2y\right)\left(2x+y\right)+7\left(2x+y\right)^{2}$$

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

$$13x^{2}-52xy+52y^{2}-8\left(x-2y\right)\left(2x+y\right)+7\left(4x^{2}+4xy+y^{2}\right)$$

Use the distributive property to multiply 7 by 4x^{2}+4xy+y^{2}.

$$13x^{2}-52xy+52y^{2}-8\left(x-2y\right)\left(2x+y\right)+28x^{2}+28xy+7y^{2}$$

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

$$13x^{2}-52xy+52y^{2}+\left(-8x+16y\right)\left(2x+y\right)+28x^{2}+28xy+7y^{2}$$

Use the distributive property to multiply -8x+16y by 2x+y and combine like terms.

$$13x^{2}-52xy+52y^{2}-16x^{2}+24xy+16y^{2}+28x^{2}+28xy+7y^{2}$$

Combine 13x^{2} and -16x^{2} to get -3x^{2}.

$$-3x^{2}-52xy+52y^{2}+24xy+16y^{2}+28x^{2}+28xy+7y^{2}$$

Combine -52xy and 24xy to get -28xy.

$$-3x^{2}-28xy+52y^{2}+16y^{2}+28x^{2}+28xy+7y^{2}$$

Combine 52y^{2} and 16y^{2} to get 68y^{2}.

$$-3x^{2}-28xy+68y^{2}+28x^{2}+28xy+7y^{2}$$

Combine -3x^{2} and 28x^{2} to get 25x^{2}.

$$25x^{2}-28xy+68y^{2}+28xy+7y^{2}$$

Combine -28xy and 28xy to get 0.

$$25x^{2}+68y^{2}+7y^{2}$$

Combine 68y^{2} and 7y^{2} to get 75y^{2}.

$$25x^{2}+75y^{2}$$