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

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

Baholash

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

Yechim qadamlari

\frac{1}{4}r-s+\frac{2}{3}t kvadratini chiqarish.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(r+\frac{1}{4}s\right)^{2} kengaytirilishi uchun ishlating.

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

r^{2}+\frac{1}{2}rs+\frac{1}{16}s^{2} teskarisini topish uchun har birining teskarisini toping.

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

-\frac{15}{16}r^{2} ni olish uchun \frac{1}{16}r^{2} va -r^{2} ni birlashtirish.

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

-rs ni olish uchun -\frac{1}{2}rs va -\frac{1}{2}rs ni birlashtirish.

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

\frac{15}{16}s^{2} ni olish uchun s^{2} va -\frac{1}{16}s^{2} ni birlashtirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(s-\frac{2}{3}t\right)^{2} kengaytirilishi uchun ishlating.

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

s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2} teskarisini topish uchun har birining teskarisini toping.

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

-\frac{1}{16}s^{2} ni olish uchun \frac{15}{16}s^{2} va -s^{2} ni birlashtirish.

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

0 ni olish uchun -\frac{4}{3}st va \frac{4}{3}st ni birlashtirish.

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

0 ni olish uchun \frac{4}{9}t^{2} va -\frac{4}{9}t^{2} ni birlashtirish.

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

\frac{1}{16} ga r+s ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{1}{16}r+\frac{1}{16}s ga 15r+s ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

0 ni olish uchun -\frac{15}{16}r^{2} va \frac{15}{16}r^{2} ni birlashtirish.

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

0 ni olish uchun -rs va rs ni birlashtirish.

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

0 ni olish uchun -\frac{1}{16}s^{2} va \frac{1}{16}s^{2} ni birlashtirish.

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

Kengaytirish

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

Yechim qadamlari

\frac{1}{4}r-s+\frac{2}{3}t kvadratini chiqarish.

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

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(r+\frac{1}{4}s\right)^{2} kengaytirilishi uchun ishlating.

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

r^{2}+\frac{1}{2}rs+\frac{1}{16}s^{2} teskarisini topish uchun har birining teskarisini toping.

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

-\frac{15}{16}r^{2} ni olish uchun \frac{1}{16}r^{2} va -r^{2} ni birlashtirish.

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

-rs ni olish uchun -\frac{1}{2}rs va -\frac{1}{2}rs ni birlashtirish.

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

\frac{15}{16}s^{2} ni olish uchun s^{2} va -\frac{1}{16}s^{2} ni birlashtirish.

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

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(s-\frac{2}{3}t\right)^{2} kengaytirilishi uchun ishlating.

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

s^{2}-\frac{4}{3}st+\frac{4}{9}t^{2} teskarisini topish uchun har birining teskarisini toping.

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

-\frac{1}{16}s^{2} ni olish uchun \frac{15}{16}s^{2} va -s^{2} ni birlashtirish.

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

0 ni olish uchun -\frac{4}{3}st va \frac{4}{3}st ni birlashtirish.

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

0 ni olish uchun \frac{4}{9}t^{2} va -\frac{4}{9}t^{2} ni birlashtirish.

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

\frac{1}{16} ga r+s ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

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

\frac{1}{16}r+\frac{1}{16}s ga 15r+s ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

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

0 ni olish uchun -\frac{15}{16}r^{2} va \frac{15}{16}r^{2} ni birlashtirish.

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

0 ni olish uchun -rs va rs ni birlashtirish.

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

0 ni olish uchun -\frac{1}{16}s^{2} va \frac{1}{16}s^{2} ni birlashtirish.

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