Solve 4(a+b)^2-12(a^2-b^2)+9(a-b)^2

$$4 ( a + b ) ^ { 2 } - 12 ( a ^ { 2 } - b ^ { 2 } ) + 9 ( a - b ) ^ { 2 }$$

Evaluate

$$\left(a-5b\right)^{2}$$

Solution Steps

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

$$4\left(a^{2}+2ab+b^{2}\right)-12\left(a^{2}-b^{2}\right)+9\left(a-b\right)^{2}$$

Use the distributive property to multiply 4 by a^{2}+2ab+b^{2}.

$$4a^{2}+8ab+4b^{2}-12\left(a^{2}-b^{2}\right)+9\left(a-b\right)^{2}$$

Use the distributive property to multiply -12 by a^{2}-b^{2}.

$$4a^{2}+8ab+4b^{2}-12a^{2}+12b^{2}+9\left(a-b\right)^{2}$$

Combine 4a^{2} and -12a^{2} to get -8a^{2}.

$$-8a^{2}+8ab+4b^{2}+12b^{2}+9\left(a-b\right)^{2}$$

Combine 4b^{2} and 12b^{2} to get 16b^{2}.

$$-8a^{2}+8ab+16b^{2}+9\left(a-b\right)^{2}$$

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

$$-8a^{2}+8ab+16b^{2}+9\left(a^{2}-2ab+b^{2}\right)$$

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

$$-8a^{2}+8ab+16b^{2}+9a^{2}-18ab+9b^{2}$$

Combine -8a^{2} and 9a^{2} to get a^{2}.

$$a^{2}+8ab+16b^{2}-18ab+9b^{2}$$

Combine 8ab and -18ab to get -10ab.

$$a^{2}-10ab+16b^{2}+9b^{2}$$

Combine 16b^{2} and 9b^{2} to get 25b^{2}.

$$a^{2}-10ab+25b^{2}$$

Expand

$$a^{2}-10ab+25b^{2}$$

Solution Steps

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

$$4\left(a^{2}+2ab+b^{2}\right)-12\left(a^{2}-b^{2}\right)+9\left(a-b\right)^{2}$$

Use the distributive property to multiply 4 by a^{2}+2ab+b^{2}.

$$4a^{2}+8ab+4b^{2}-12\left(a^{2}-b^{2}\right)+9\left(a-b\right)^{2}$$

Use the distributive property to multiply -12 by a^{2}-b^{2}.

$$4a^{2}+8ab+4b^{2}-12a^{2}+12b^{2}+9\left(a-b\right)^{2}$$

Combine 4a^{2} and -12a^{2} to get -8a^{2}.

$$-8a^{2}+8ab+4b^{2}+12b^{2}+9\left(a-b\right)^{2}$$

Combine 4b^{2} and 12b^{2} to get 16b^{2}.

$$-8a^{2}+8ab+16b^{2}+9\left(a-b\right)^{2}$$

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

$$-8a^{2}+8ab+16b^{2}+9\left(a^{2}-2ab+b^{2}\right)$$

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

$$-8a^{2}+8ab+16b^{2}+9a^{2}-18ab+9b^{2}$$

Combine -8a^{2} and 9a^{2} to get a^{2}.

$$a^{2}+8ab+16b^{2}-18ab+9b^{2}$$

Combine 8ab and -18ab to get -10ab.

$$a^{2}-10ab+16b^{2}+9b^{2}$$

Combine 16b^{2} and 9b^{2} to get 25b^{2}.

$$a^{2}-10ab+25b^{2}$$