(16-2t)^2=5t^2+36 eching

$$( 16 - 2 t ) ^ { 2 } = 5 t ^ { 2 } + 36$$

t uchun yechish

$$t=2\sqrt{311}-32\approx 3,270384177$$

$$t=-2\sqrt{311}-32\approx -67,270384177$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(16-2t\right)^{2} kengaytirilishi uchun ishlating.

$$256-64t+4t^{2}=5t^{2}+36$$

Ikkala tarafdan 5t^{2} ni ayirish.

$$256-64t+4t^{2}-5t^{2}=36$$

-t^{2} ni olish uchun 4t^{2} va -5t^{2} ni birlashtirish.

$$256-64t-t^{2}=36$$

Ikkala tarafdan 36 ni ayirish.

$$256-64t-t^{2}-36=0$$

220 olish uchun 256 dan 36 ni ayirish.

$$220-64t-t^{2}=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$-t^{2}-64t+220=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -64 ni b va 220 ni c bilan almashtiring.

$$t=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\left(-1\right)\times 220}}{2\left(-1\right)}$$

-64 kvadratini chiqarish.

$$t=\frac{-\left(-64\right)±\sqrt{4096-4\left(-1\right)\times 220}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$t=\frac{-\left(-64\right)±\sqrt{4096+4\times 220}}{2\left(-1\right)}$$

4 ni 220 marotabaga ko'paytirish.

$$t=\frac{-\left(-64\right)±\sqrt{4096+880}}{2\left(-1\right)}$$

4096 ni 880 ga qo'shish.

$$t=\frac{-\left(-64\right)±\sqrt{4976}}{2\left(-1\right)}$$

4976 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-64\right)±4\sqrt{311}}{2\left(-1\right)}$$

-64 ning teskarisi 64 ga teng.

$$t=\frac{64±4\sqrt{311}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$t=\frac{64±4\sqrt{311}}{-2}$$

t=\frac{64±4\sqrt{311}}{-2} tenglamasini yeching, bunda ± musbat. 64 ni 4\sqrt{311} ga qo'shish.

$$t=\frac{4\sqrt{311}+64}{-2}$$

64+4\sqrt{311} ni -2 ga bo'lish.

$$t=-2\sqrt{311}-32$$

t=\frac{64±4\sqrt{311}}{-2} tenglamasini yeching, bunda ± manfiy. 64 dan 4\sqrt{311} ni ayirish.

$$t=\frac{64-4\sqrt{311}}{-2}$$

64-4\sqrt{311} ni -2 ga bo'lish.

$$t=2\sqrt{311}-32$$

Tenglama yechildi.

$$t=-2\sqrt{311}-32 t=2\sqrt{311}-32$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(16-2t\right)^{2} kengaytirilishi uchun ishlating.

$$256-64t+4t^{2}=5t^{2}+36$$

Ikkala tarafdan 5t^{2} ni ayirish.

$$256-64t+4t^{2}-5t^{2}=36$$

-t^{2} ni olish uchun 4t^{2} va -5t^{2} ni birlashtirish.

$$256-64t-t^{2}=36$$

Ikkala tarafdan 256 ni ayirish.

$$-64t-t^{2}=36-256$$

-220 olish uchun 36 dan 256 ni ayirish.

$$-64t-t^{2}=-220$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-t^{2}-64t=-220$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-t^{2}-64t}{-1}=-\frac{220}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$t^{2}+\left(-\frac{64}{-1}\right)t=-\frac{220}{-1}$$

-64 ni -1 ga bo'lish.

$$t^{2}+64t=-\frac{220}{-1}$$

-220 ni -1 ga bo'lish.

$$t^{2}+64t=220$$

64 ni bo‘lish, x shartining koeffitsienti, 2 ga 32 olish uchun. Keyin, 32 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}+64t+32^{2}=220+32^{2}$$

32 kvadratini chiqarish.

$$t^{2}+64t+1024=220+1024$$

220 ni 1024 ga qo'shish.

$$t^{2}+64t+1024=1244$$

t^{2}+64t+1024 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t+32\right)^{2}=1244$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t+32\right)^{2}}=\sqrt{1244}$$

Qisqartirish.

$$t+32=2\sqrt{311} t+32=-2\sqrt{311}$$

Tenglamaning ikkala tarafidan 32 ni ayirish.

$$t=2\sqrt{311}-32 t=-2\sqrt{311}-32$$