-16t^2+92t+20=0 eching

$$- 16 t ^ { 2 } + 92 t + 20 = 0$$

t uchun yechish

$$t = \frac{\sqrt{609} + 23}{8} \approx 5,95974067$$

$$t=\frac{23-\sqrt{609}}{8}\approx -0,20974067$$

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.

$$-16t^{2}+92t+20=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -16 ni a, 92 ni b va 20 ni c bilan almashtiring.

$$t=\frac{-92±\sqrt{92^{2}-4\left(-16\right)\times 20}}{2\left(-16\right)}$$

92 kvadratini chiqarish.

$$t=\frac{-92±\sqrt{8464-4\left(-16\right)\times 20}}{2\left(-16\right)}$$

-4 ni -16 marotabaga ko'paytirish.

$$t=\frac{-92±\sqrt{8464+64\times 20}}{2\left(-16\right)}$$

64 ni 20 marotabaga ko'paytirish.

$$t=\frac{-92±\sqrt{8464+1280}}{2\left(-16\right)}$$

8464 ni 1280 ga qo'shish.

$$t=\frac{-92±\sqrt{9744}}{2\left(-16\right)}$$

9744 ning kvadrat ildizini chiqarish.

$$t=\frac{-92±4\sqrt{609}}{2\left(-16\right)}$$

2 ni -16 marotabaga ko'paytirish.

$$t=\frac{-92±4\sqrt{609}}{-32}$$

t=\frac{-92±4\sqrt{609}}{-32} tenglamasini yeching, bunda ± musbat. -92 ni 4\sqrt{609} ga qo'shish.

$$t=\frac{4\sqrt{609}-92}{-32}$$

-92+4\sqrt{609} ni -32 ga bo'lish.

$$t=\frac{23-\sqrt{609}}{8}$$

t=\frac{-92±4\sqrt{609}}{-32} tenglamasini yeching, bunda ± manfiy. -92 dan 4\sqrt{609} ni ayirish.

$$t=\frac{-4\sqrt{609}-92}{-32}$$

-92-4\sqrt{609} ni -32 ga bo'lish.

$$t=\frac{\sqrt{609}+23}{8}$$

Tenglama yechildi.

$$t=\frac{23-\sqrt{609}}{8} t=\frac{\sqrt{609}+23}{8}$$

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.

$$-16t^{2}+92t+20=0$$

Tenglamaning ikkala tarafidan 20 ni ayirish.

$$-16t^{2}+92t+20-20=-20$$

O‘zidan 20 ayirilsa 0 qoladi.

$$-16t^{2}+92t=-20$$

Ikki tarafini -16 ga bo‘ling.

$$\frac{-16t^{2}+92t}{-16}=-\frac{20}{-16}$$

-16 ga bo'lish -16 ga ko'paytirishni bekor qiladi.

$$t^{2}+\frac{92}{-16}t=-\frac{20}{-16}$$

\frac{92}{-16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}-\frac{23}{4}t=-\frac{20}{-16}$$

\frac{-20}{-16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}-\frac{23}{4}t=\frac{5}{4}$$

-\frac{23}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{23}{8} olish uchun. Keyin, -\frac{23}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{23}{4}t+\left(-\frac{23}{8}\right)^{2}=\frac{5}{4}+\left(-\frac{23}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{23}{8} kvadratini chiqarish.

$$t^{2}-\frac{23}{4}t+\frac{529}{64}=\frac{5}{4}+\frac{529}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{4} ni \frac{529}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{23}{4}t+\frac{529}{64}=\frac{609}{64}$$

t^{2}-\frac{23}{4}t+\frac{529}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(t-\frac{23}{8}\right)^{2}=\frac{609}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{23}{8}\right)^{2}}=\sqrt{\frac{609}{64}}$$

Qisqartirish.

$$t-\frac{23}{8}=\frac{\sqrt{609}}{8} t-\frac{23}{8}=-\frac{\sqrt{609}}{8}$$

\frac{23}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{609}+23}{8} t=\frac{23-\sqrt{609}}{8}$$