Solve N(x)=(3x-5y)^2-2(3x-5y)(7y-11x)+(7y-71x)^2

$$N ( x ) = ( 3 x - 5 y ) ^ { 2 } - 2 ( 3 x - 5 y ) ( 7 y - 11 x ) + ( 7 y - 71 x ) ^ { 2 }$$

Evaluate

$$5116x^{2}-1176xy+144y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-5y\right)^{2}.

$$9x^{2}-30xy+25y^{2}-2\left(3x-5y\right)\left(7y-11x\right)+\left(7y-71x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7y-71x\right)^{2}.

$$9x^{2}-30xy+25y^{2}-2\left(3x-5y\right)\left(7y-11x\right)+49y^{2}-994yx+5041x^{2}$$

Use the distributive property to multiply -2 by 3x-5y.

$$9x^{2}-30xy+25y^{2}+\left(-6x+10y\right)\left(7y-11x\right)+49y^{2}-994yx+5041x^{2}$$

Use the distributive property to multiply -6x+10y by 7y-11x and combine like terms.

$$9x^{2}-30xy+25y^{2}-152xy+66x^{2}+70y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine -30xy and -152xy to get -182xy.

$$9x^{2}-182xy+25y^{2}+66x^{2}+70y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine 9x^{2} and 66x^{2} to get 75x^{2}.

$$75x^{2}-182xy+25y^{2}+70y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine 25y^{2} and 70y^{2} to get 95y^{2}.

$$75x^{2}-182xy+95y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine 95y^{2} and 49y^{2} to get 144y^{2}.

$$75x^{2}-182xy+144y^{2}-994yx+5041x^{2}$$

Combine -182xy and -994yx to get -1176xy.

$$75x^{2}-1176xy+144y^{2}+5041x^{2}$$

Combine 75x^{2} and 5041x^{2} to get 5116x^{2}.

$$5116x^{2}-1176xy+144y^{2}$$

Expand

$$5116x^{2}-1176xy+144y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3x-5y\right)^{2}.

$$9x^{2}-30xy+25y^{2}-2\left(3x-5y\right)\left(7y-11x\right)+\left(7y-71x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(7y-71x\right)^{2}.

$$9x^{2}-30xy+25y^{2}-2\left(3x-5y\right)\left(7y-11x\right)+49y^{2}-994yx+5041x^{2}$$

Use the distributive property to multiply -2 by 3x-5y.

$$9x^{2}-30xy+25y^{2}+\left(-6x+10y\right)\left(7y-11x\right)+49y^{2}-994yx+5041x^{2}$$

Use the distributive property to multiply -6x+10y by 7y-11x and combine like terms.

$$9x^{2}-30xy+25y^{2}-152xy+66x^{2}+70y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine -30xy and -152xy to get -182xy.

$$9x^{2}-182xy+25y^{2}+66x^{2}+70y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine 9x^{2} and 66x^{2} to get 75x^{2}.

$$75x^{2}-182xy+25y^{2}+70y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine 25y^{2} and 70y^{2} to get 95y^{2}.

$$75x^{2}-182xy+95y^{2}+49y^{2}-994yx+5041x^{2}$$

Combine 95y^{2} and 49y^{2} to get 144y^{2}.

$$75x^{2}-182xy+144y^{2}-994yx+5041x^{2}$$

Combine -182xy and -994yx to get -1176xy.

$$75x^{2}-1176xy+144y^{2}+5041x^{2}$$

Combine 75x^{2} and 5041x^{2} to get 5116x^{2}.

$$5116x^{2}-1176xy+144y^{2}$$