Solve (a-3+b)^2+(a-b)^2-2(a^2+b^2)+6(a-b)

$$( a - 3 + b ) ^ { 2 } + ( a - b ) ^ { 2 } - 2 ( a ^ { 2 } + b ^ { 2 } ) + 6 ( a - b )$$

Evaluate

$$9-12b$$

Solution Steps

Square a-3+b.

$$a^{2}+2ab-6a+b^{2}-6b+9+\left(a-b\right)^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$a^{2}+2ab-6a+b^{2}-6b+9+a^{2}-2ab+b^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$2a^{2}+2ab-6a+b^{2}-6b+9-2ab+b^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Combine 2ab and -2ab to get 0.

$$2a^{2}-6a+b^{2}-6b+9+b^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$2a^{2}-6a+2b^{2}-6b+9-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Use the distributive property to multiply -2 by a^{2}+b^{2}.

$$2a^{2}-6a+2b^{2}-6b+9-2a^{2}-2b^{2}+6\left(a-b\right)$$

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

$$-6a+2b^{2}-6b+9-2b^{2}+6\left(a-b\right)$$

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

$$-6a-6b+9+6\left(a-b\right)$$

Use the distributive property to multiply 6 by a-b.

$$-6a-6b+9+6a-6b$$

Combine -6a and 6a to get 0.

$$-6b+9-6b$$

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

$$-12b+9$$

Expand

$$9-12b$$

Solution Steps

Square a-3+b.

$$a^{2}+2ab-6a+b^{2}-6b+9+\left(a-b\right)^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(a-b\right)^{2}.

$$a^{2}+2ab-6a+b^{2}-6b+9+a^{2}-2ab+b^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Combine a^{2} and a^{2} to get 2a^{2}.

$$2a^{2}+2ab-6a+b^{2}-6b+9-2ab+b^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Combine 2ab and -2ab to get 0.

$$2a^{2}-6a+b^{2}-6b+9+b^{2}-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$2a^{2}-6a+2b^{2}-6b+9-2\left(a^{2}+b^{2}\right)+6\left(a-b\right)$$

Use the distributive property to multiply -2 by a^{2}+b^{2}.

$$2a^{2}-6a+2b^{2}-6b+9-2a^{2}-2b^{2}+6\left(a-b\right)$$

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

$$-6a+2b^{2}-6b+9-2b^{2}+6\left(a-b\right)$$

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

$$-6a-6b+9+6\left(a-b\right)$$

Use the distributive property to multiply 6 by a-b.

$$-6a-6b+9+6a-6b$$

Combine -6a and 6a to get 0.

$$-6b+9-6b$$

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

$$-12b+9$$