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

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

Calcular

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

Pasos da solución

Usar teorema binomial \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para expandir \left(3+\lambda \right)^{2}.

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

Usa a propiedade distributiva para multiplicar \frac{5}{2} por 9+6\lambda +\lambda ^{2}.

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

Usar teorema binomial \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para expandir \left(9\lambda -3\right)^{2}.

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

Usa a propiedade distributiva para multiplicar -\frac{5}{6} por 81\lambda ^{2}-54\lambda +9.

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

Combina \frac{5}{2}\lambda ^{2} e -\frac{135}{2}\lambda ^{2} para obter -65\lambda ^{2}.

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

Combina 15\lambda e 45\lambda para obter 60\lambda .

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

Resta \frac{15}{2} de \frac{45}{2} para obter 15.

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

Usar teorema binomial \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para expandir \left(6-3\lambda \right)^{2}.

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

Usa a propiedade distributiva para multiplicar \frac{5}{3} por 36-36\lambda +9\lambda ^{2}.

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

Suma 15 e 60 para obter 75.

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

Combina 60\lambda e -60\lambda para obter 0.

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

Combina -65\lambda ^{2} e 15\lambda ^{2} para obter -50\lambda ^{2}.

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

Expandir

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

Pasos da solución

Usar teorema binomial \left(a+b\right)^{2}=a^{2}+2ab+b^{2} para expandir \left(3+\lambda \right)^{2}.

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

Usa a propiedade distributiva para multiplicar \frac{5}{2} por 9+6\lambda +\lambda ^{2}.

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

Usar teorema binomial \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para expandir \left(9\lambda -3\right)^{2}.

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

Usa a propiedade distributiva para multiplicar -\frac{5}{6} por 81\lambda ^{2}-54\lambda +9.

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

Combina \frac{5}{2}\lambda ^{2} e -\frac{135}{2}\lambda ^{2} para obter -65\lambda ^{2}.

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

Combina 15\lambda e 45\lambda para obter 60\lambda .

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

Resta \frac{15}{2} de \frac{45}{2} para obter 15.

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

Usar teorema binomial \left(a-b\right)^{2}=a^{2}-2ab+b^{2} para expandir \left(6-3\lambda \right)^{2}.

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

Usa a propiedade distributiva para multiplicar \frac{5}{3} por 36-36\lambda +9\lambda ^{2}.

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

Suma 15 e 60 para obter 75.

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

Combina 60\lambda e -60\lambda para obter 0.

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

Combina -65\lambda ^{2} e 15\lambda ^{2} para obter -50\lambda ^{2}.

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