Solve left(5x-7yright)^2-(2x-y)*(2x+y)-left(3x+4yright)^2

$${ \left(5x-7y \right) }^{ 2 } -(2x-y) \times (2x+y)- { \left(3x+4y \right) }^{ 2 }$$

Evaluate

$$12x^{2}-94xy+34y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-7y\right)^{2}.

$$25x^{2}-70xy+49y^{2}-\left(2x-y\right)\left(2x+y\right)-\left(3x+4y\right)^{2}$$

Consider \left(2x-y\right)\left(2x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$25x^{2}-70xy+49y^{2}-\left(\left(2x\right)^{2}-y^{2}\right)-\left(3x+4y\right)^{2}$$

Expand \left(2x\right)^{2}.

$$25x^{2}-70xy+49y^{2}-\left(2^{2}x^{2}-y^{2}\right)-\left(3x+4y\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-70xy+49y^{2}-\left(4x^{2}-y^{2}\right)-\left(3x+4y\right)^{2}$$

To find the opposite of 4x^{2}-y^{2}, find the opposite of each term.

$$25x^{2}-70xy+49y^{2}-4x^{2}+y^{2}-\left(3x+4y\right)^{2}$$

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

$$21x^{2}-70xy+49y^{2}+y^{2}-\left(3x+4y\right)^{2}$$

Combine 49y^{2} and y^{2} to get 50y^{2}.

$$21x^{2}-70xy+50y^{2}-\left(3x+4y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+4y\right)^{2}.

$$21x^{2}-70xy+50y^{2}-\left(9x^{2}+24xy+16y^{2}\right)$$

To find the opposite of 9x^{2}+24xy+16y^{2}, find the opposite of each term.

$$21x^{2}-70xy+50y^{2}-9x^{2}-24xy-16y^{2}$$

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

$$12x^{2}-70xy+50y^{2}-24xy-16y^{2}$$

Combine -70xy and -24xy to get -94xy.

$$12x^{2}-94xy+50y^{2}-16y^{2}$$

Combine 50y^{2} and -16y^{2} to get 34y^{2}.

$$12x^{2}-94xy+34y^{2}$$

Expand

$$12x^{2}-94xy+34y^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5x-7y\right)^{2}.

$$25x^{2}-70xy+49y^{2}-\left(2x-y\right)\left(2x+y\right)-\left(3x+4y\right)^{2}$$

Consider \left(2x-y\right)\left(2x+y\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$25x^{2}-70xy+49y^{2}-\left(\left(2x\right)^{2}-y^{2}\right)-\left(3x+4y\right)^{2}$$

Expand \left(2x\right)^{2}.

$$25x^{2}-70xy+49y^{2}-\left(2^{2}x^{2}-y^{2}\right)-\left(3x+4y\right)^{2}$$

Calculate 2 to the power of 2 and get 4.

$$25x^{2}-70xy+49y^{2}-\left(4x^{2}-y^{2}\right)-\left(3x+4y\right)^{2}$$

To find the opposite of 4x^{2}-y^{2}, find the opposite of each term.

$$25x^{2}-70xy+49y^{2}-4x^{2}+y^{2}-\left(3x+4y\right)^{2}$$

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

$$21x^{2}-70xy+49y^{2}+y^{2}-\left(3x+4y\right)^{2}$$

Combine 49y^{2} and y^{2} to get 50y^{2}.

$$21x^{2}-70xy+50y^{2}-\left(3x+4y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3x+4y\right)^{2}.

$$21x^{2}-70xy+50y^{2}-\left(9x^{2}+24xy+16y^{2}\right)$$

To find the opposite of 9x^{2}+24xy+16y^{2}, find the opposite of each term.

$$21x^{2}-70xy+50y^{2}-9x^{2}-24xy-16y^{2}$$

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

$$12x^{2}-70xy+50y^{2}-24xy-16y^{2}$$

Combine -70xy and -24xy to get -94xy.

$$12x^{2}-94xy+50y^{2}-16y^{2}$$

Combine 50y^{2} and -16y^{2} to get 34y^{2}.

$$12x^{2}-94xy+34y^{2}$$