7t^2-5t-9=0 eching

$$7 t ^ { 2 } - 5 t - 9 = 0$$

t uchun yechish

$$t = \frac{\sqrt{277} + 5}{14} \approx 1,545951213$$

$$t=\frac{5-\sqrt{277}}{14}\approx -0,831665498$$

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.

$$7t^{2}-5t-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, -5 ni b va -9 ni c bilan almashtiring.

$$t=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 7\left(-9\right)}}{2\times 7}$$

-5 kvadratini chiqarish.

$$t=\frac{-\left(-5\right)±\sqrt{25-4\times 7\left(-9\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$t=\frac{-\left(-5\right)±\sqrt{25-28\left(-9\right)}}{2\times 7}$$

-28 ni -9 marotabaga ko'paytirish.

$$t=\frac{-\left(-5\right)±\sqrt{25+252}}{2\times 7}$$

25 ni 252 ga qo'shish.

$$t=\frac{-\left(-5\right)±\sqrt{277}}{2\times 7}$$

-5 ning teskarisi 5 ga teng.

$$t=\frac{5±\sqrt{277}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$t=\frac{5±\sqrt{277}}{14}$$

t=\frac{5±\sqrt{277}}{14} tenglamasini yeching, bunda ± musbat. 5 ni \sqrt{277} ga qo'shish.

$$t=\frac{\sqrt{277}+5}{14}$$

t=\frac{5±\sqrt{277}}{14} tenglamasini yeching, bunda ± manfiy. 5 dan \sqrt{277} ni ayirish.

$$t=\frac{5-\sqrt{277}}{14}$$

Tenglama yechildi.

$$t=\frac{\sqrt{277}+5}{14} t=\frac{5-\sqrt{277}}{14}$$

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.

$$7t^{2}-5t-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$7t^{2}-5t-9-\left(-9\right)=-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$7t^{2}-5t=-\left(-9\right)$$

0 dan -9 ni ayirish.

$$7t^{2}-5t=9$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7t^{2}-5t}{7}=\frac{9}{7}$$

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

$$t^{2}-\frac{5}{7}t=\frac{9}{7}$$

-\frac{5}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{14} olish uchun. Keyin, -\frac{5}{14} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{5}{7}t+\left(-\frac{5}{14}\right)^{2}=\frac{9}{7}+\left(-\frac{5}{14}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{14} kvadratini chiqarish.

$$t^{2}-\frac{5}{7}t+\frac{25}{196}=\frac{9}{7}+\frac{25}{196}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{7} ni \frac{25}{196} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{5}{7}t+\frac{25}{196}=\frac{277}{196}$$

t^{2}-\frac{5}{7}t+\frac{25}{196} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{5}{14}\right)^{2}=\frac{277}{196}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{5}{14}\right)^{2}}=\sqrt{\frac{277}{196}}$$

Qisqartirish.

$$t-\frac{5}{14}=\frac{\sqrt{277}}{14} t-\frac{5}{14}=-\frac{\sqrt{277}}{14}$$

\frac{5}{14} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{277}+5}{14} t=\frac{5-\sqrt{277}}{14}$$