57/16t^2-85/16t=-250 eching

$$\frac { 57 } { 16 } t ^ { 2 } - \frac { 85 } { 16 } t = - 250$$

t uchun yechish

$$t=\frac{85+5\sqrt{36191}i}{114}\approx 0,745614035+8,343829954i$$

$$t=\frac{-5\sqrt{36191}i+85}{114}\approx 0,745614035-8,343829954i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$\frac{57}{16}t^{2}-\frac{85}{16}t=-250$$

250 ni tenglamaning ikkala tarafiga qo'shish.

$$\frac{57}{16}t^{2}-\frac{85}{16}t-\left(-250\right)=-250-\left(-250\right)$$

O‘zidan -250 ayirilsa 0 qoladi.

$$\frac{57}{16}t^{2}-\frac{85}{16}t-\left(-250\right)=0$$

0 dan -250 ni ayirish.

$$\frac{57}{16}t^{2}-\frac{85}{16}t+250=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{57}{16} ni a, -\frac{85}{16} ni b va 250 ni c bilan almashtiring.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\left(-\frac{85}{16}\right)^{2}-4\times \frac{57}{16}\times 250}}{2\times \frac{57}{16}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{85}{16} kvadratini chiqarish.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-4\times \frac{57}{16}\times 250}}{2\times \frac{57}{16}}$$

-4 ni \frac{57}{16} marotabaga ko'paytirish.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-\frac{57}{4}\times 250}}{2\times \frac{57}{16}}$$

-\frac{57}{4} ni 250 marotabaga ko'paytirish.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{\frac{7225}{256}-\frac{7125}{2}}}{2\times \frac{57}{16}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7225}{256} ni -\frac{7125}{2} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t=\frac{-\left(-\frac{85}{16}\right)±\sqrt{-\frac{904775}{256}}}{2\times \frac{57}{16}}$$

-\frac{904775}{256} ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-\frac{85}{16}\right)±\frac{5\sqrt{36191}i}{16}}{2\times \frac{57}{16}}$$

-\frac{85}{16} ning teskarisi \frac{85}{16} ga teng.

$$t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{2\times \frac{57}{16}}$$

2 ni \frac{57}{16} marotabaga ko'paytirish.

$$t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}}$$

t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}} tenglamasini yeching, bunda ± musbat. \frac{85}{16} ni \frac{5i\sqrt{36191}}{16} ga qo'shish.

$$t=\frac{85+5\sqrt{36191}i}{\frac{57}{8}\times 16}$$

\frac{85+5i\sqrt{36191}}{16} ni \frac{57}{8} ga bo'lish \frac{85+5i\sqrt{36191}}{16} ga k'paytirish \frac{57}{8} ga qaytarish.

$$t=\frac{85+5\sqrt{36191}i}{114}$$

t=\frac{\frac{85}{16}±\frac{5\sqrt{36191}i}{16}}{\frac{57}{8}} tenglamasini yeching, bunda ± manfiy. \frac{85}{16} dan \frac{5i\sqrt{36191}}{16} ni ayirish.

$$t=\frac{-5\sqrt{36191}i+85}{\frac{57}{8}\times 16}$$

\frac{85-5i\sqrt{36191}}{16} ni \frac{57}{8} ga bo'lish \frac{85-5i\sqrt{36191}}{16} ga k'paytirish \frac{57}{8} ga qaytarish.

$$t=\frac{-5\sqrt{36191}i+85}{114}$$

Tenglama yechildi.

$$t=\frac{85+5\sqrt{36191}i}{114} t=\frac{-5\sqrt{36191}i+85}{114}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$\frac{57}{16}t^{2}-\frac{85}{16}t=-250$$

Tenglamaning ikki tarafini \frac{57}{16} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{\frac{57}{16}t^{2}-\frac{85}{16}t}{\frac{57}{16}}=-\frac{250}{\frac{57}{16}}$$

\frac{57}{16} ga bo'lish \frac{57}{16} ga ko'paytirishni bekor qiladi.

$$t^{2}+\left(-\frac{\frac{85}{16}}{\frac{57}{16}}\right)t=-\frac{250}{\frac{57}{16}}$$

-\frac{85}{16} ni \frac{57}{16} ga bo'lish -\frac{85}{16} ga k'paytirish \frac{57}{16} ga qaytarish.

$$t^{2}-\frac{85}{57}t=-\frac{250}{\frac{57}{16}}$$

-250 ni \frac{57}{16} ga bo'lish -250 ga k'paytirish \frac{57}{16} ga qaytarish.

$$t^{2}-\frac{85}{57}t=-\frac{4000}{57}$$

-\frac{85}{57} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{85}{114} olish uchun. Keyin, -\frac{85}{114} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{85}{57}t+\left(-\frac{85}{114}\right)^{2}=-\frac{4000}{57}+\left(-\frac{85}{114}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{85}{114} kvadratini chiqarish.

$$t^{2}-\frac{85}{57}t+\frac{7225}{12996}=-\frac{4000}{57}+\frac{7225}{12996}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{4000}{57} ni \frac{7225}{12996} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{85}{57}t+\frac{7225}{12996}=-\frac{904775}{12996}$$

t^{2}-\frac{85}{57}t+\frac{7225}{12996} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{85}{114}\right)^{2}=-\frac{904775}{12996}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{85}{114}\right)^{2}}=\sqrt{-\frac{904775}{12996}}$$

Qisqartirish.

$$t-\frac{85}{114}=\frac{5\sqrt{36191}i}{114} t-\frac{85}{114}=-\frac{5\sqrt{36191}i}{114}$$

\frac{85}{114} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{85+5\sqrt{36191}i}{114} t=\frac{-5\sqrt{36191}i+85}{114}$$