Solve 5/2(3+lambda)^2-5/6(9lambda-3)^2+(5/3)(6-3lambda)^2=

$$\frac { 5 } { 2 } ( 3 + \lambda ) ^ { 2 } - \frac { 5 } { 6 } ( 9 \lambda - 3 ) ^ { 2 } + ( \frac { 5 } { 3 } ) ( 6 - 3 \lambda ) ^ { 2 } =$$

Evaluate

$$75-50\lambda ^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\lambda \right)^{2}.

$$\frac{5}{2}\left(9+6\lambda +\lambda ^{2}\right)-\frac{5}{6}\left(9\lambda -3\right)^{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use the distributive property to multiply \frac{5}{2} by 9+6\lambda +\lambda ^{2}.

$$\frac{45}{2}+15\lambda +\frac{5}{2}\lambda ^{2}-\frac{5}{6}\left(9\lambda -3\right)^{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9\lambda -3\right)^{2}.

$$\frac{45}{2}+15\lambda +\frac{5}{2}\lambda ^{2}-\frac{5}{6}\left(81\lambda ^{2}-54\lambda +9\right)+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use the distributive property to multiply -\frac{5}{6} by 81\lambda ^{2}-54\lambda +9.

$$\frac{45}{2}+15\lambda +\frac{5}{2}\lambda ^{2}-\frac{135}{2}\lambda ^{2}+45\lambda -\frac{15}{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Combine \frac{5}{2}\lambda ^{2} and -\frac{135}{2}\lambda ^{2} to get -65\lambda ^{2}.

$$\frac{45}{2}+15\lambda -65\lambda ^{2}+45\lambda -\frac{15}{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Combine 15\lambda and 45\lambda to get 60\lambda .

$$\frac{45}{2}+60\lambda -65\lambda ^{2}-\frac{15}{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Subtract \frac{15}{2} from \frac{45}{2} to get 15.

$$15+60\lambda -65\lambda ^{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-3\lambda \right)^{2}.

$$15+60\lambda -65\lambda ^{2}+\frac{5}{3}\left(36-36\lambda +9\lambda ^{2}\right)$$

Use the distributive property to multiply \frac{5}{3} by 36-36\lambda +9\lambda ^{2}.

$$15+60\lambda -65\lambda ^{2}+60-60\lambda +15\lambda ^{2}$$

Add 15 and 60 to get 75.

$$75+60\lambda -65\lambda ^{2}-60\lambda +15\lambda ^{2}$$

Combine 60\lambda and -60\lambda to get 0.

$$75-65\lambda ^{2}+15\lambda ^{2}$$

Combine -65\lambda ^{2} and 15\lambda ^{2} to get -50\lambda ^{2}.

$$75-50\lambda ^{2}$$

Expand

$$75-50\lambda ^{2}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\lambda \right)^{2}.

$$\frac{5}{2}\left(9+6\lambda +\lambda ^{2}\right)-\frac{5}{6}\left(9\lambda -3\right)^{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use the distributive property to multiply \frac{5}{2} by 9+6\lambda +\lambda ^{2}.

$$\frac{45}{2}+15\lambda +\frac{5}{2}\lambda ^{2}-\frac{5}{6}\left(9\lambda -3\right)^{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(9\lambda -3\right)^{2}.

$$\frac{45}{2}+15\lambda +\frac{5}{2}\lambda ^{2}-\frac{5}{6}\left(81\lambda ^{2}-54\lambda +9\right)+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use the distributive property to multiply -\frac{5}{6} by 81\lambda ^{2}-54\lambda +9.

$$\frac{45}{2}+15\lambda +\frac{5}{2}\lambda ^{2}-\frac{135}{2}\lambda ^{2}+45\lambda -\frac{15}{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Combine \frac{5}{2}\lambda ^{2} and -\frac{135}{2}\lambda ^{2} to get -65\lambda ^{2}.

$$\frac{45}{2}+15\lambda -65\lambda ^{2}+45\lambda -\frac{15}{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Combine 15\lambda and 45\lambda to get 60\lambda .

$$\frac{45}{2}+60\lambda -65\lambda ^{2}-\frac{15}{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Subtract \frac{15}{2} from \frac{45}{2} to get 15.

$$15+60\lambda -65\lambda ^{2}+\frac{5}{3}\left(6-3\lambda \right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(6-3\lambda \right)^{2}.

$$15+60\lambda -65\lambda ^{2}+\frac{5}{3}\left(36-36\lambda +9\lambda ^{2}\right)$$

Use the distributive property to multiply \frac{5}{3} by 36-36\lambda +9\lambda ^{2}.

$$15+60\lambda -65\lambda ^{2}+60-60\lambda +15\lambda ^{2}$$

Add 15 and 60 to get 75.

$$75+60\lambda -65\lambda ^{2}-60\lambda +15\lambda ^{2}$$

Combine 60\lambda and -60\lambda to get 0.

$$75-65\lambda ^{2}+15\lambda ^{2}$$

Combine -65\lambda ^{2} and 15\lambda ^{2} to get -50\lambda ^{2}.

$$75-50\lambda ^{2}$$