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

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

Evaluate

$$17x^{2}+22xy-23y^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by 9x^{2}+6xy+y^{2}.

$$18x^{2}+12xy+2y^{2}-\left(x-5y\right)^{2}$$

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

$$18x^{2}+12xy+2y^{2}-\left(x^{2}-10xy+25y^{2}\right)$$

To find the opposite of x^{2}-10xy+25y^{2}, find the opposite of each term.

$$18x^{2}+12xy+2y^{2}-x^{2}+10xy-25y^{2}$$

Combine 18x^{2} and -x^{2} to get 17x^{2}.

$$17x^{2}+12xy+2y^{2}+10xy-25y^{2}$$

Combine 12xy and 10xy to get 22xy.

$$17x^{2}+22xy+2y^{2}-25y^{2}$$

Combine 2y^{2} and -25y^{2} to get -23y^{2}.

$$17x^{2}+22xy-23y^{2}$$

Expand

$$17x^{2}+22xy-23y^{2}$$

Solution Steps

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

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

Use the distributive property to multiply 2 by 9x^{2}+6xy+y^{2}.

$$18x^{2}+12xy+2y^{2}-\left(x-5y\right)^{2}$$

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

$$18x^{2}+12xy+2y^{2}-\left(x^{2}-10xy+25y^{2}\right)$$

To find the opposite of x^{2}-10xy+25y^{2}, find the opposite of each term.

$$18x^{2}+12xy+2y^{2}-x^{2}+10xy-25y^{2}$$

Combine 18x^{2} and -x^{2} to get 17x^{2}.

$$17x^{2}+12xy+2y^{2}+10xy-25y^{2}$$

Combine 12xy and 10xy to get 22xy.

$$17x^{2}+22xy+2y^{2}-25y^{2}$$

Combine 2y^{2} and -25y^{2} to get -23y^{2}.

$$17x^{2}+22xy-23y^{2}$$