Solve (3x-y)^2-(3x+y)^2+5x(y-x)

$$( 3 x - y ) ^ { 2 } - ( 3 x + y ) ^ { 2 } + 5 x ( y - x )$$

Evaluate

$$-x\left(5x+7y\right)$$

Solution Steps

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

$$9x^{2}-6xy+y^{2}-\left(3x+y\right)^{2}+5x\left(y-x\right)$$

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

$$9x^{2}-6xy+y^{2}-\left(9x^{2}+6xy+y^{2}\right)+5x\left(y-x\right)$$

To find the opposite of 9x^{2}+6xy+y^{2}, find the opposite of each term.

$$9x^{2}-6xy+y^{2}-9x^{2}-6xy-y^{2}+5x\left(y-x\right)$$

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

$$-6xy+y^{2}-6xy-y^{2}+5x\left(y-x\right)$$

Combine -6xy and -6xy to get -12xy.

$$-12xy+y^{2}-y^{2}+5x\left(y-x\right)$$

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

$$-12xy+5x\left(y-x\right)$$

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

$$-12xy+5xy-5x^{2}$$

Combine -12xy and 5xy to get -7xy.

$$-7xy-5x^{2}$$

Expand

$$-5x^{2}-7xy$$

Solution Steps

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

$$9x^{2}-6xy+y^{2}-\left(3x+y\right)^{2}+5x\left(y-x\right)$$

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

$$9x^{2}-6xy+y^{2}-\left(9x^{2}+6xy+y^{2}\right)+5x\left(y-x\right)$$

To find the opposite of 9x^{2}+6xy+y^{2}, find the opposite of each term.

$$9x^{2}-6xy+y^{2}-9x^{2}-6xy-y^{2}+5x\left(y-x\right)$$

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

$$-6xy+y^{2}-6xy-y^{2}+5x\left(y-x\right)$$

Combine -6xy and -6xy to get -12xy.

$$-12xy+y^{2}-y^{2}+5x\left(y-x\right)$$

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

$$-12xy+5x\left(y-x\right)$$

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

$$-12xy+5xy-5x^{2}$$

Combine -12xy and 5xy to get -7xy.

$$-7xy-5x^{2}$$