Solve (13x+4y)^2-(25x+6y)^2+(16y-3x)^2-(14x+8y)^2=

$$( 13 x + 4 y ) ^ { 2 } - ( 25 x + 6 y ) ^ { 2 } + ( 16 y - 3 x ) ^ { 2 } - ( 14 x + 8 y ) ^ { 2 } =$$

Evaluate

$$172y^{2}-516xy-643x^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(13x+4y\right)^{2}.

$$169x^{2}+104xy+16y^{2}-\left(25x+6y\right)^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(25x+6y\right)^{2}.

$$169x^{2}+104xy+16y^{2}-\left(625x^{2}+300xy+36y^{2}\right)+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

To find the opposite of 625x^{2}+300xy+36y^{2}, find the opposite of each term.

$$169x^{2}+104xy+16y^{2}-625x^{2}-300xy-36y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Combine 169x^{2} and -625x^{2} to get -456x^{2}.

$$-456x^{2}+104xy+16y^{2}-300xy-36y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Combine 104xy and -300xy to get -196xy.

$$-456x^{2}-196xy+16y^{2}-36y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Combine 16y^{2} and -36y^{2} to get -20y^{2}.

$$-456x^{2}-196xy-20y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16y-3x\right)^{2}.

$$-456x^{2}-196xy-20y^{2}+256y^{2}-96yx+9x^{2}-\left(14x+8y\right)^{2}$$

Combine -20y^{2} and 256y^{2} to get 236y^{2}.

$$-456x^{2}-196xy+236y^{2}-96yx+9x^{2}-\left(14x+8y\right)^{2}$$

Combine -196xy and -96yx to get -292xy.

$$-456x^{2}-292xy+236y^{2}+9x^{2}-\left(14x+8y\right)^{2}$$

Combine -456x^{2} and 9x^{2} to get -447x^{2}.

$$-447x^{2}-292xy+236y^{2}-\left(14x+8y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(14x+8y\right)^{2}.

$$-447x^{2}-292xy+236y^{2}-\left(196x^{2}+224xy+64y^{2}\right)$$

To find the opposite of 196x^{2}+224xy+64y^{2}, find the opposite of each term.

$$-447x^{2}-292xy+236y^{2}-196x^{2}-224xy-64y^{2}$$

Combine -447x^{2} and -196x^{2} to get -643x^{2}.

$$-643x^{2}-292xy+236y^{2}-224xy-64y^{2}$$

Combine -292xy and -224xy to get -516xy.

$$-643x^{2}-516xy+236y^{2}-64y^{2}$$

Combine 236y^{2} and -64y^{2} to get 172y^{2}.

$$-643x^{2}-516xy+172y^{2}$$

Expand

$$172y^{2}-516xy-643x^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(13x+4y\right)^{2}.

$$169x^{2}+104xy+16y^{2}-\left(25x+6y\right)^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(25x+6y\right)^{2}.

$$169x^{2}+104xy+16y^{2}-\left(625x^{2}+300xy+36y^{2}\right)+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

To find the opposite of 625x^{2}+300xy+36y^{2}, find the opposite of each term.

$$169x^{2}+104xy+16y^{2}-625x^{2}-300xy-36y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Combine 169x^{2} and -625x^{2} to get -456x^{2}.

$$-456x^{2}+104xy+16y^{2}-300xy-36y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Combine 104xy and -300xy to get -196xy.

$$-456x^{2}-196xy+16y^{2}-36y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Combine 16y^{2} and -36y^{2} to get -20y^{2}.

$$-456x^{2}-196xy-20y^{2}+\left(16y-3x\right)^{2}-\left(14x+8y\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16y-3x\right)^{2}.

$$-456x^{2}-196xy-20y^{2}+256y^{2}-96yx+9x^{2}-\left(14x+8y\right)^{2}$$

Combine -20y^{2} and 256y^{2} to get 236y^{2}.

$$-456x^{2}-196xy+236y^{2}-96yx+9x^{2}-\left(14x+8y\right)^{2}$$

Combine -196xy and -96yx to get -292xy.

$$-456x^{2}-292xy+236y^{2}+9x^{2}-\left(14x+8y\right)^{2}$$

Combine -456x^{2} and 9x^{2} to get -447x^{2}.

$$-447x^{2}-292xy+236y^{2}-\left(14x+8y\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(14x+8y\right)^{2}.

$$-447x^{2}-292xy+236y^{2}-\left(196x^{2}+224xy+64y^{2}\right)$$

To find the opposite of 196x^{2}+224xy+64y^{2}, find the opposite of each term.

$$-447x^{2}-292xy+236y^{2}-196x^{2}-224xy-64y^{2}$$

Combine -447x^{2} and -196x^{2} to get -643x^{2}.

$$-643x^{2}-292xy+236y^{2}-224xy-64y^{2}$$

Combine -292xy and -224xy to get -516xy.

$$-643x^{2}-516xy+236y^{2}-64y^{2}$$

Combine 236y^{2} and -64y^{2} to get 172y^{2}.

$$-643x^{2}-516xy+172y^{2}$$