47=73t-5t^2 eching

$$47 = 73 t - 5 t ^ { 2 }$$

t uchun yechish

$$t = \frac{\sqrt{4389} + 73}{10} \approx 13,92495283$$

$$t=\frac{73-\sqrt{4389}}{10}\approx 0,67504717$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$73t-5t^{2}=47$$

Ikkala tarafdan 47 ni ayirish.

$$73t-5t^{2}-47=0$$

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.

$$-5t^{2}+73t-47=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, 73 ni b va -47 ni c bilan almashtiring.

$$t=\frac{-73±\sqrt{73^{2}-4\left(-5\right)\left(-47\right)}}{2\left(-5\right)}$$

73 kvadratini chiqarish.

$$t=\frac{-73±\sqrt{5329-4\left(-5\right)\left(-47\right)}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$t=\frac{-73±\sqrt{5329+20\left(-47\right)}}{2\left(-5\right)}$$

20 ni -47 marotabaga ko'paytirish.

$$t=\frac{-73±\sqrt{5329-940}}{2\left(-5\right)}$$

5329 ni -940 ga qo'shish.

$$t=\frac{-73±\sqrt{4389}}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$t=\frac{-73±\sqrt{4389}}{-10}$$

t=\frac{-73±\sqrt{4389}}{-10} tenglamasini yeching, bunda ± musbat. -73 ni \sqrt{4389} ga qo'shish.

$$t=\frac{\sqrt{4389}-73}{-10}$$

-73+\sqrt{4389} ni -10 ga bo'lish.

$$t=\frac{73-\sqrt{4389}}{10}$$

t=\frac{-73±\sqrt{4389}}{-10} tenglamasini yeching, bunda ± manfiy. -73 dan \sqrt{4389} ni ayirish.

$$t=\frac{-\sqrt{4389}-73}{-10}$$

-73-\sqrt{4389} ni -10 ga bo'lish.

$$t=\frac{\sqrt{4389}+73}{10}$$

Tenglama yechildi.

$$t=\frac{73-\sqrt{4389}}{10} t=\frac{\sqrt{4389}+73}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$73t-5t^{2}=47$$

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

$$-5t^{2}+73t=47$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5t^{2}+73t}{-5}=\frac{47}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$t^{2}+\frac{73}{-5}t=\frac{47}{-5}$$

73 ni -5 ga bo'lish.

$$t^{2}-\frac{73}{5}t=\frac{47}{-5}$$

47 ni -5 ga bo'lish.

$$t^{2}-\frac{73}{5}t=-\frac{47}{5}$$

-\frac{73}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{73}{10} olish uchun. Keyin, -\frac{73}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{73}{5}t+\left(-\frac{73}{10}\right)^{2}=-\frac{47}{5}+\left(-\frac{73}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{73}{10} kvadratini chiqarish.

$$t^{2}-\frac{73}{5}t+\frac{5329}{100}=-\frac{47}{5}+\frac{5329}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{47}{5} ni \frac{5329}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{73}{5}t+\frac{5329}{100}=\frac{4389}{100}$$

t^{2}-\frac{73}{5}t+\frac{5329}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{73}{10}\right)^{2}=\frac{4389}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{73}{10}\right)^{2}}=\sqrt{\frac{4389}{100}}$$

Qisqartirish.

$$t-\frac{73}{10}=\frac{\sqrt{4389}}{10} t-\frac{73}{10}=-\frac{\sqrt{4389}}{10}$$

\frac{73}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{4389}+73}{10} t=\frac{73-\sqrt{4389}}{10}$$