Solve 81left(a+bright)^2-36left(2a^2+5ab+2b^2right)

$$81 { \left( a+b \right) }^{ 2 } -36 \left( 2 { a }^{ 2 } +5ab+2 { b }^{ 2 } \right)$$

Evaluate

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

Solution Steps

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

$$81\left(a^{2}+2ab+b^{2}\right)-36\left(2a^{2}+5ab+2b^{2}\right)$$

Use the distributive property to multiply 81 by a^{2}+2ab+b^{2}.

$$81a^{2}+162ab+81b^{2}-36\left(2a^{2}+5ab+2b^{2}\right)$$

Use the distributive property to multiply -36 by 2a^{2}+5ab+2b^{2}.

$$81a^{2}+162ab+81b^{2}-72a^{2}-180ab-72b^{2}$$

Combine 81a^{2} and -72a^{2} to get 9a^{2}.

$$9a^{2}+162ab+81b^{2}-180ab-72b^{2}$$

Combine 162ab and -180ab to get -18ab.

$$9a^{2}-18ab+81b^{2}-72b^{2}$$

Combine 81b^{2} and -72b^{2} to get 9b^{2}.

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

Expand

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

Solution Steps

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

$$81\left(a^{2}+2ab+b^{2}\right)-36\left(2a^{2}+5ab+2b^{2}\right)$$

Use the distributive property to multiply 81 by a^{2}+2ab+b^{2}.

$$81a^{2}+162ab+81b^{2}-36\left(2a^{2}+5ab+2b^{2}\right)$$

Use the distributive property to multiply -36 by 2a^{2}+5ab+2b^{2}.

$$81a^{2}+162ab+81b^{2}-72a^{2}-180ab-72b^{2}$$

Combine 81a^{2} and -72a^{2} to get 9a^{2}.

$$9a^{2}+162ab+81b^{2}-180ab-72b^{2}$$

Combine 162ab and -180ab to get -18ab.

$$9a^{2}-18ab+81b^{2}-72b^{2}$$

Combine 81b^{2} and -72b^{2} to get 9b^{2}.

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