Solve (3ab-cd)^2+9(cd-ab)^2+6(3ab-cd)(cd-ab)

$$( 3 a b - c d ) ^ { 2 } + 9 ( c d - a b ) ^ { 2 } + 6 ( 3 a b - c d ) ( c d - a b )$$

Evaluate

$$4\left(cd\right)^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(3ab-cd\right)^{2}.

$$9a^{2}b^{2}-6abcd+c^{2}d^{2}+9\left(cd-ab\right)^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

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

$$9a^{2}b^{2}-6abcd+c^{2}d^{2}+9\left(c^{2}d^{2}-2cdab+a^{2}b^{2}\right)+6\left(3ab-cd\right)\left(cd-ab\right)$$

Use the distributive property to multiply 9 by c^{2}d^{2}-2cdab+a^{2}b^{2}.

$$9a^{2}b^{2}-6abcd+c^{2}d^{2}+9c^{2}d^{2}-18dcab+9a^{2}b^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Combine c^{2}d^{2} and 9c^{2}d^{2} to get 10c^{2}d^{2}.

$$9a^{2}b^{2}-6abcd+10c^{2}d^{2}-18dcab+9a^{2}b^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Combine -6abcd and -18dcab to get -24abcd.

$$9a^{2}b^{2}-24abcd+10c^{2}d^{2}+9a^{2}b^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Combine 9a^{2}b^{2} and 9a^{2}b^{2} to get 18a^{2}b^{2}.

$$18a^{2}b^{2}-24abcd+10c^{2}d^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Use the distributive property to multiply 6 by 3ab-cd.

$$18a^{2}b^{2}-24abcd+10c^{2}d^{2}+\left(18ab-6dc\right)\left(cd-ab\right)$$

Use the distributive property to multiply 18ab-6dc by cd-ab and combine like terms.

$$18a^{2}b^{2}-24abcd+10c^{2}d^{2}+24abcd-18a^{2}b^{2}-6d^{2}c^{2}$$

Combine -24abcd and 24abcd to get 0.

$$18a^{2}b^{2}+10c^{2}d^{2}-18a^{2}b^{2}-6d^{2}c^{2}$$

Combine 18a^{2}b^{2} and -18a^{2}b^{2} to get 0.

$$10c^{2}d^{2}-6d^{2}c^{2}$$

Combine 10c^{2}d^{2} and -6d^{2}c^{2} to get 4c^{2}d^{2}.

$$4c^{2}d^{2}$$

Expand

$$4\left(cd\right)^{2}$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(3ab-cd\right)^{2}.

$$9a^{2}b^{2}-6abcd+c^{2}d^{2}+9\left(cd-ab\right)^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

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

$$9a^{2}b^{2}-6abcd+c^{2}d^{2}+9\left(c^{2}d^{2}-2cdab+a^{2}b^{2}\right)+6\left(3ab-cd\right)\left(cd-ab\right)$$

Use the distributive property to multiply 9 by c^{2}d^{2}-2cdab+a^{2}b^{2}.

$$9a^{2}b^{2}-6abcd+c^{2}d^{2}+9c^{2}d^{2}-18dcab+9a^{2}b^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Combine c^{2}d^{2} and 9c^{2}d^{2} to get 10c^{2}d^{2}.

$$9a^{2}b^{2}-6abcd+10c^{2}d^{2}-18dcab+9a^{2}b^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Combine -6abcd and -18dcab to get -24abcd.

$$9a^{2}b^{2}-24abcd+10c^{2}d^{2}+9a^{2}b^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Combine 9a^{2}b^{2} and 9a^{2}b^{2} to get 18a^{2}b^{2}.

$$18a^{2}b^{2}-24abcd+10c^{2}d^{2}+6\left(3ab-cd\right)\left(cd-ab\right)$$

Use the distributive property to multiply 6 by 3ab-cd.

$$18a^{2}b^{2}-24abcd+10c^{2}d^{2}+\left(18ab-6dc\right)\left(cd-ab\right)$$

Use the distributive property to multiply 18ab-6dc by cd-ab and combine like terms.

$$18a^{2}b^{2}-24abcd+10c^{2}d^{2}+24abcd-18a^{2}b^{2}-6d^{2}c^{2}$$

Combine -24abcd and 24abcd to get 0.

$$18a^{2}b^{2}+10c^{2}d^{2}-18a^{2}b^{2}-6d^{2}c^{2}$$

Combine 18a^{2}b^{2} and -18a^{2}b^{2} to get 0.

$$10c^{2}d^{2}-6d^{2}c^{2}$$

Combine 10c^{2}d^{2} and -6d^{2}c^{2} to get 4c^{2}d^{2}.

$$4c^{2}d^{2}$$