Resol (1/4r-s+2/3t)^2-(r+1/4s)^2-(s-2/3t)^2+frac{1}{16}(r+s)(15r+s)

$$( \frac { 1 } { 4 } r - s + \frac { 2 } { 3 } t ) ^ { 2 } - ( r + \frac { 1 } { 4 } s ) ^ { 2 } - ( s - \frac { 2 } { 3 } t ) ^ { 2 } + \frac { 1 } { 16 } ( r + s ) ( 15 r + s )$$

Calcula

$$\frac{rt}{3}$$

Passos de solució

Eleveu \frac{1}{4}r-s+\frac{2}{3}t al quadrat.

$$\frac{1}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(r+\frac{1}{4}s\right)^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Utilitzeu el teorema del binomi \left(a+b\right)^{2}=a^{2}+2ab+b^{2} per desenvolupar \left(r+\frac{1}{4}s\right)^{2}.

$$\frac{1}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(r^{2}+\frac{1}{2}rs+\frac{1}{16}s^{2}\right)-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Per trobar l'oposat de r^{2}+\frac{1}{2}rs+\frac{1}{16}s^{2}, cerqueu l'oposat de cada terme.

$$\frac{1}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-r^{2}-\frac{1}{2}rs-\frac{1}{16}s^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu \frac{1}{16}r^{2} i -r^{2} per obtenir -\frac{15}{16}r^{2}.

$$-\frac{15}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\frac{1}{2}rs-\frac{1}{16}s^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu -\frac{1}{2}rs i -\frac{1}{2}rs per obtenir -rs.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\frac{1}{16}s^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu s^{2} i -\frac{1}{16}s^{2} per obtenir \frac{15}{16}s^{2}.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+\frac{15}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Utilitzeu el teorema del binomi \left(a-b\right)^{2}=a^{2}-2ab+b^{2} per desenvolupar \left(s-\frac{2}{3}t\right)^{2}.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+\frac{15}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}\right)+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Per trobar l'oposat de s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}, cerqueu l'oposat de cada terme.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+\frac{15}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-s^{2}+\frac{4}{3}st-\frac{4}{9}t^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu \frac{15}{16}s^{2} i -s^{2} per obtenir -\frac{1}{16}s^{2}.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}+\frac{4}{3}st-\frac{4}{9}t^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu -\frac{4}{3}st i \frac{4}{3}st per obtenir 0.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{4}{9}t^{2}-\frac{4}{9}t^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu \frac{4}{9}t^{2} i -\frac{4}{9}t^{2} per obtenir 0.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Utilitzeu la propietat distributiva per multiplicar \frac{1}{16} per r+s.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\left(\frac{1}{16}r+\frac{1}{16}s\right)\left(15r+s\right)$$

Utilitzeu la propietat distributiva per multiplicar \frac{1}{16}r+\frac{1}{16}s per 15r+s i combinar-los com termes.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{15}{16}r^{2}+rs+\frac{1}{16}s^{2}$$

Combineu -\frac{15}{16}r^{2} i \frac{15}{16}r^{2} per obtenir 0.

$$-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+rs+\frac{1}{16}s^{2}$$

Combineu -rs i rs per obtenir 0.

$$\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{1}{16}s^{2}$$

Combineu -\frac{1}{16}s^{2} i \frac{1}{16}s^{2} per obtenir 0.

$$\frac{1}{3}rt$$

Expandiu

$$\frac{rt}{3}$$

Passos de solució

Eleveu \frac{1}{4}r-s+\frac{2}{3}t al quadrat.

$$\frac{1}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(r+\frac{1}{4}s\right)^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Utilitzeu el teorema del binomi \left(a+b\right)^{2}=a^{2}+2ab+b^{2} per desenvolupar \left(r+\frac{1}{4}s\right)^{2}.

$$\frac{1}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(r^{2}+\frac{1}{2}rs+\frac{1}{16}s^{2}\right)-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Per trobar l'oposat de r^{2}+\frac{1}{2}rs+\frac{1}{16}s^{2}, cerqueu l'oposat de cada terme.

$$\frac{1}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-r^{2}-\frac{1}{2}rs-\frac{1}{16}s^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu \frac{1}{16}r^{2} i -r^{2} per obtenir -\frac{15}{16}r^{2}.

$$-\frac{15}{16}r^{2}-\frac{1}{2}rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\frac{1}{2}rs-\frac{1}{16}s^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu -\frac{1}{2}rs i -\frac{1}{2}rs per obtenir -rs.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\frac{1}{16}s^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu s^{2} i -\frac{1}{16}s^{2} per obtenir \frac{15}{16}s^{2}.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+\frac{15}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(s-\frac{2}{3}t\right)^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Utilitzeu el teorema del binomi \left(a-b\right)^{2}=a^{2}-2ab+b^{2} per desenvolupar \left(s-\frac{2}{3}t\right)^{2}.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+\frac{15}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-\left(s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}\right)+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Per trobar l'oposat de s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}, cerqueu l'oposat de cada terme.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt+\frac{15}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}-s^{2}+\frac{4}{3}st-\frac{4}{9}t^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu \frac{15}{16}s^{2} i -s^{2} per obtenir -\frac{1}{16}s^{2}.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2}+\frac{4}{3}st-\frac{4}{9}t^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu -\frac{4}{3}st i \frac{4}{3}st per obtenir 0.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{4}{9}t^{2}-\frac{4}{9}t^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Combineu \frac{4}{9}t^{2} i -\frac{4}{9}t^{2} per obtenir 0.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{1}{16}\left(r+s\right)\left(15r+s\right)$$

Utilitzeu la propietat distributiva per multiplicar \frac{1}{16} per r+s.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\left(\frac{1}{16}r+\frac{1}{16}s\right)\left(15r+s\right)$$

Utilitzeu la propietat distributiva per multiplicar \frac{1}{16}r+\frac{1}{16}s per 15r+s i combinar-los com termes.

$$-\frac{15}{16}r^{2}-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{15}{16}r^{2}+rs+\frac{1}{16}s^{2}$$

Combineu -\frac{15}{16}r^{2} i \frac{15}{16}r^{2} per obtenir 0.

$$-rs+\frac{1}{3}rt-\frac{1}{16}s^{2}+rs+\frac{1}{16}s^{2}$$

Combineu -rs i rs per obtenir 0.

$$\frac{1}{3}rt-\frac{1}{16}s^{2}+\frac{1}{16}s^{2}$$

Combineu -\frac{1}{16}s^{2} i \frac{1}{16}s^{2} per obtenir 0.

$$\frac{1}{3}rt$$