7t^2-32t+12=0 eching

$$7 t ^ { 2 } - 32 t + 12 = 0$$

t uchun yechish

$$t = \frac{2 \sqrt{43} + 16}{7} \approx 4,15926815$$

$$t=\frac{16-2\sqrt{43}}{7}\approx 0,412160422$$

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.

$$7t^{2}-32t+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -32 ni b va 12 ni c bilan almashtiring.

$$t=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\times 7\times 12}}{2\times 7}$$

-32 kvadratini chiqarish.

$$t=\frac{-\left(-32\right)±\sqrt{1024-4\times 7\times 12}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$t=\frac{-\left(-32\right)±\sqrt{1024-28\times 12}}{2\times 7}$$

-28 ni 12 marotabaga ko'paytirish.

$$t=\frac{-\left(-32\right)±\sqrt{1024-336}}{2\times 7}$$

1024 ni -336 ga qo'shish.

$$t=\frac{-\left(-32\right)±\sqrt{688}}{2\times 7}$$

688 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-32\right)±4\sqrt{43}}{2\times 7}$$

-32 ning teskarisi 32 ga teng.

$$t=\frac{32±4\sqrt{43}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$t=\frac{32±4\sqrt{43}}{14}$$

t=\frac{32±4\sqrt{43}}{14} tenglamasini yeching, bunda ± musbat. 32 ni 4\sqrt{43} ga qo'shish.

$$t=\frac{4\sqrt{43}+32}{14}$$

32+4\sqrt{43} ni 14 ga bo'lish.

$$t=\frac{2\sqrt{43}+16}{7}$$

t=\frac{32±4\sqrt{43}}{14} tenglamasini yeching, bunda ± manfiy. 32 dan 4\sqrt{43} ni ayirish.

$$t=\frac{32-4\sqrt{43}}{14}$$

32-4\sqrt{43} ni 14 ga bo'lish.

$$t=\frac{16-2\sqrt{43}}{7}$$

Tenglama yechildi.

$$t=\frac{2\sqrt{43}+16}{7} t=\frac{16-2\sqrt{43}}{7}$$

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.

$$7t^{2}-32t+12=0$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$7t^{2}-32t+12-12=-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$7t^{2}-32t=-12$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7t^{2}-32t}{7}=-\frac{12}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$t^{2}-\frac{32}{7}t=-\frac{12}{7}$$

-\frac{32}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{16}{7} olish uchun. Keyin, -\frac{16}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{32}{7}t+\left(-\frac{16}{7}\right)^{2}=-\frac{12}{7}+\left(-\frac{16}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{16}{7} kvadratini chiqarish.

$$t^{2}-\frac{32}{7}t+\frac{256}{49}=-\frac{12}{7}+\frac{256}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{12}{7} ni \frac{256}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{32}{7}t+\frac{256}{49}=\frac{172}{49}$$

t^{2}-\frac{32}{7}t+\frac{256}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{16}{7}\right)^{2}=\frac{172}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{16}{7}\right)^{2}}=\sqrt{\frac{172}{49}}$$

Qisqartirish.

$$t-\frac{16}{7}=\frac{2\sqrt{43}}{7} t-\frac{16}{7}=-\frac{2\sqrt{43}}{7}$$

\frac{16}{7} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{2\sqrt{43}+16}{7} t=\frac{16-2\sqrt{43}}{7}$$