Solve (5x-y)^2-(y+4x)^2-9x^2

$$( 5 x - y ) ^ { 2 } - ( y + 4 x ) ^ { 2 } - 9 x ^ { 2 }$$

Evaluate

$$-18xy$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-y\right)^{2}.

$$25x^{2}-10xy+y^{2}-\left(y+4x\right)^{2}-9x^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+4x\right)^{2}.

$$25x^{2}-10xy+y^{2}-\left(y^{2}+8yx+16x^{2}\right)-9x^{2}$$

To find the opposite of y^{2}+8yx+16x^{2}, find the opposite of each term.

$$25x^{2}-10xy+y^{2}-y^{2}-8yx-16x^{2}-9x^{2}$$

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

$$25x^{2}-10xy-8yx-16x^{2}-9x^{2}$$

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

$$25x^{2}-18xy-16x^{2}-9x^{2}$$

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

$$9x^{2}-18xy-9x^{2}$$

Combine 9x^{2} and -9x^{2} to get 0.

$$-18xy$$

Expand

$$-18xy$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-y\right)^{2}.

$$25x^{2}-10xy+y^{2}-\left(y+4x\right)^{2}-9x^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+4x\right)^{2}.

$$25x^{2}-10xy+y^{2}-\left(y^{2}+8yx+16x^{2}\right)-9x^{2}$$

To find the opposite of y^{2}+8yx+16x^{2}, find the opposite of each term.

$$25x^{2}-10xy+y^{2}-y^{2}-8yx-16x^{2}-9x^{2}$$

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

$$25x^{2}-10xy-8yx-16x^{2}-9x^{2}$$

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

$$25x^{2}-18xy-16x^{2}-9x^{2}$$

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

$$9x^{2}-18xy-9x^{2}$$

Combine 9x^{2} and -9x^{2} to get 0.

$$-18xy$$