Solve 9left(x-yright)^2-12left(y^2-x^2right)+4left(x+yright)^2

$$9 { \left(x-y \right) }^{ 2 } -12 \left( { y }^{ 2 } - { x }^{ 2 } \right) +4 { \left(x+y \right) }^{ 2 }$$

Evaluate

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

Solution Steps

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

$$9\left(x^{2}-2xy+y^{2}\right)-12\left(y^{2}-x^{2}\right)+4\left(x+y\right)^{2}$$

Use the distributive property to multiply 9 by x^{2}-2xy+y^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(y^{2}-x^{2}\right)+4\left(x+y\right)^{2}$$

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

$$9x^{2}-18xy+9y^{2}-12y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine 9y^{2} and -12y^{2} to get -3y^{2}.

$$9x^{2}-18xy-3y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine 9x^{2} and 12x^{2} to get 21x^{2}.

$$21x^{2}-18xy-3y^{2}+4\left(x+y\right)^{2}$$

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

$$21x^{2}-18xy-3y^{2}+4\left(x^{2}+2xy+y^{2}\right)$$

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

$$21x^{2}-18xy-3y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 21x^{2} and 4x^{2} to get 25x^{2}.

$$25x^{2}-18xy-3y^{2}+8xy+4y^{2}$$

Combine -18xy and 8xy to get -10xy.

$$25x^{2}-10xy-3y^{2}+4y^{2}$$

Combine -3y^{2} and 4y^{2} to get y^{2}.

$$25x^{2}-10xy+y^{2}$$

Expand

$$25x^{2}-10xy+y^{2}$$

Solution Steps

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

$$9\left(x^{2}-2xy+y^{2}\right)-12\left(y^{2}-x^{2}\right)+4\left(x+y\right)^{2}$$

Use the distributive property to multiply 9 by x^{2}-2xy+y^{2}.

$$9x^{2}-18xy+9y^{2}-12\left(y^{2}-x^{2}\right)+4\left(x+y\right)^{2}$$

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

$$9x^{2}-18xy+9y^{2}-12y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine 9y^{2} and -12y^{2} to get -3y^{2}.

$$9x^{2}-18xy-3y^{2}+12x^{2}+4\left(x+y\right)^{2}$$

Combine 9x^{2} and 12x^{2} to get 21x^{2}.

$$21x^{2}-18xy-3y^{2}+4\left(x+y\right)^{2}$$

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

$$21x^{2}-18xy-3y^{2}+4\left(x^{2}+2xy+y^{2}\right)$$

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

$$21x^{2}-18xy-3y^{2}+4x^{2}+8xy+4y^{2}$$

Combine 21x^{2} and 4x^{2} to get 25x^{2}.

$$25x^{2}-18xy-3y^{2}+8xy+4y^{2}$$

Combine -18xy and 8xy to get -10xy.

$$25x^{2}-10xy-3y^{2}+4y^{2}$$

Combine -3y^{2} and 4y^{2} to get y^{2}.

$$25x^{2}-10xy+y^{2}$$