5t^2-72t-108=0 eching

$$5 t ^ { 2 } - 72 t - 108 = 0$$

t uchun yechish

$$t = \frac{6 \sqrt{51} + 36}{5} \approx 15,769714114$$

$$t=\frac{36-6\sqrt{51}}{5}\approx -1,369714114$$

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.

$$5t^{2}-72t-108=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -72 ni b va -108 ni c bilan almashtiring.

$$t=\frac{-\left(-72\right)±\sqrt{\left(-72\right)^{2}-4\times 5\left(-108\right)}}{2\times 5}$$

-72 kvadratini chiqarish.

$$t=\frac{-\left(-72\right)±\sqrt{5184-4\times 5\left(-108\right)}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$t=\frac{-\left(-72\right)±\sqrt{5184-20\left(-108\right)}}{2\times 5}$$

-20 ni -108 marotabaga ko'paytirish.

$$t=\frac{-\left(-72\right)±\sqrt{5184+2160}}{2\times 5}$$

5184 ni 2160 ga qo'shish.

$$t=\frac{-\left(-72\right)±\sqrt{7344}}{2\times 5}$$

7344 ning kvadrat ildizini chiqarish.

$$t=\frac{-\left(-72\right)±12\sqrt{51}}{2\times 5}$$

-72 ning teskarisi 72 ga teng.

$$t=\frac{72±12\sqrt{51}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$t=\frac{72±12\sqrt{51}}{10}$$

t=\frac{72±12\sqrt{51}}{10} tenglamasini yeching, bunda ± musbat. 72 ni 12\sqrt{51} ga qo'shish.

$$t=\frac{12\sqrt{51}+72}{10}$$

72+12\sqrt{51} ni 10 ga bo'lish.

$$t=\frac{6\sqrt{51}+36}{5}$$

t=\frac{72±12\sqrt{51}}{10} tenglamasini yeching, bunda ± manfiy. 72 dan 12\sqrt{51} ni ayirish.

$$t=\frac{72-12\sqrt{51}}{10}$$

72-12\sqrt{51} ni 10 ga bo'lish.

$$t=\frac{36-6\sqrt{51}}{5}$$

Tenglama yechildi.

$$t=\frac{6\sqrt{51}+36}{5} t=\frac{36-6\sqrt{51}}{5}$$

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.

$$5t^{2}-72t-108=0$$

108 ni tenglamaning ikkala tarafiga qo'shish.

$$5t^{2}-72t-108-\left(-108\right)=-\left(-108\right)$$

O‘zidan -108 ayirilsa 0 qoladi.

$$5t^{2}-72t=-\left(-108\right)$$

0 dan -108 ni ayirish.

$$5t^{2}-72t=108$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5t^{2}-72t}{5}=\frac{108}{5}$$

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

$$t^{2}-\frac{72}{5}t=\frac{108}{5}$$

-\frac{72}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{36}{5} olish uchun. Keyin, -\frac{36}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{72}{5}t+\left(-\frac{36}{5}\right)^{2}=\frac{108}{5}+\left(-\frac{36}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{36}{5} kvadratini chiqarish.

$$t^{2}-\frac{72}{5}t+\frac{1296}{25}=\frac{108}{5}+\frac{1296}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{108}{5} ni \frac{1296}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{72}{5}t+\frac{1296}{25}=\frac{1836}{25}$$

t^{2}-\frac{72}{5}t+\frac{1296}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{36}{5}\right)^{2}=\frac{1836}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{36}{5}\right)^{2}}=\sqrt{\frac{1836}{25}}$$

Qisqartirish.

$$t-\frac{36}{5}=\frac{6\sqrt{51}}{5} t-\frac{36}{5}=-\frac{6\sqrt{51}}{5}$$

\frac{36}{5} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{6\sqrt{51}+36}{5} t=\frac{36-6\sqrt{51}}{5}$$