Question

Simplify.
(1) (x − 2y +3)² + (x + 2y − 3)²
(2) (3k − 4r − 2m)² − (3k + 4r − 2m)²
(3) (7a − 6b + 5c)² + (7a + 6b − 5c)²

Answer 1

It is known that,
${\left(a+b+c\right)}^2=a^2+b^2+c^2+2ab+2bc+2ca$
$$\begin{gathered} \left(1\right){\left(x-2y+3\right)}^2+{\left(x+2y-3\right)}^2 \\ ={\left(x\right)}^2+{\left(-2y\right)}^2+{\left(3\right)}^2+2\times x\times \left(-2y\right)+2\times \left(-2y\right)\times 3+2\times 3\times x+{\left(x\right)}^2+{\left(2y\right)}^2+{\left(-3\right)}^2+2\times x\times 2y+2\times \left(2y\right)\times \left(-3\right)+2\times \left(-3\right)\times x \\ =x^2+4y^2+9-4xy-12y+6x+x^2+4y^2+9+4xy-12y-6x \\ =2x^2+8y^2+18-24y \end{gathered}$$
$$\begin{gathered} \left(2\right){\left(3k-4r-2m\right)}^2-{\left(3k+4r-2m\right)}^2 \\ ={\left(3k\right)}^2+{\left(-4r\right)}^2+{\left(-2m\right)}^2+2\times 3k\times \left(-4r\right)+2\times \left(-4r\right)\times \left(-2m\right)+2\times \left(-2m\right)\times 3k-\left\{{\left(3k\right)}^2+{\left(4r\right)}^2+{\left(-2m\right)}^2+2\times 3k\times 4r+2\times 4r\times \left(-2m\right)+2\times \left(-2m\right)\times 3k\right\} \\ =9k^2+16r^2+4m^2-24kr+16rm-12mk-9k^2-16r^2-4m^2-24kr+16rm+12mk \\ =32rm-48kr \end{gathered}$$
$$\begin{gathered} \left(3\right){\left(7a-6b+5c\right)}^2+{\left(7a+6b-5c\right)}^2 \\ ={\left(7a\right)}^2+{\left(-6b\right)}^2+{\left(5c\right)}^2+2\times 7a\times \left(-6b\right)+2\times \left(-6b\right)\times 5c+2\times 5c\times 7a+{\left(7a\right)}^2+{\left(6b\right)}^2+{\left(-5c\right)}^2+2\times 7a\times 6b+2\times 6b\times \left(-5c\right)+2\times \left(-5c\right)\times 7a \\ =49a^2+36b^2+25c^2-84ab-60bc+70ca+49a^2+36b^2+25c^2+84ab-60bc-70ca \\ =98a^2+72b^2+50c^2-120bc \end{gathered}$$