-16t^2+36t+7=0 eching

$$- 16 t ^ { 2 } + 36 t + 7 = 0$$

t uchun yechish

$$t = \frac{\sqrt{109} + 9}{8} \approx 2,430038314$$

$$t=\frac{9-\sqrt{109}}{8}\approx -0,180038314$$

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.

$$-16t^{2}+36t+7=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -16 ni a, 36 ni b va 7 ni c bilan almashtiring.

$$t=\frac{-36±\sqrt{36^{2}-4\left(-16\right)\times 7}}{2\left(-16\right)}$$

36 kvadratini chiqarish.

$$t=\frac{-36±\sqrt{1296-4\left(-16\right)\times 7}}{2\left(-16\right)}$$

-4 ni -16 marotabaga ko'paytirish.

$$t=\frac{-36±\sqrt{1296+64\times 7}}{2\left(-16\right)}$$

64 ni 7 marotabaga ko'paytirish.

$$t=\frac{-36±\sqrt{1296+448}}{2\left(-16\right)}$$

1296 ni 448 ga qo'shish.

$$t=\frac{-36±\sqrt{1744}}{2\left(-16\right)}$$

1744 ning kvadrat ildizini chiqarish.

$$t=\frac{-36±4\sqrt{109}}{2\left(-16\right)}$$

2 ni -16 marotabaga ko'paytirish.

$$t=\frac{-36±4\sqrt{109}}{-32}$$

t=\frac{-36±4\sqrt{109}}{-32} tenglamasini yeching, bunda ± musbat. -36 ni 4\sqrt{109} ga qo'shish.

$$t=\frac{4\sqrt{109}-36}{-32}$$

-36+4\sqrt{109} ni -32 ga bo'lish.

$$t=\frac{9-\sqrt{109}}{8}$$

t=\frac{-36±4\sqrt{109}}{-32} tenglamasini yeching, bunda ± manfiy. -36 dan 4\sqrt{109} ni ayirish.

$$t=\frac{-4\sqrt{109}-36}{-32}$$

-36-4\sqrt{109} ni -32 ga bo'lish.

$$t=\frac{\sqrt{109}+9}{8}$$

Tenglama yechildi.

$$t=\frac{9-\sqrt{109}}{8} t=\frac{\sqrt{109}+9}{8}$$

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.

$$-16t^{2}+36t+7=0$$

Tenglamaning ikkala tarafidan 7 ni ayirish.

$$-16t^{2}+36t+7-7=-7$$

O‘zidan 7 ayirilsa 0 qoladi.

$$-16t^{2}+36t=-7$$

Ikki tarafini -16 ga bo‘ling.

$$\frac{-16t^{2}+36t}{-16}=-\frac{7}{-16}$$

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

$$t^{2}+\frac{36}{-16}t=-\frac{7}{-16}$$

\frac{36}{-16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$t^{2}-\frac{9}{4}t=-\frac{7}{-16}$$

-7 ni -16 ga bo'lish.

$$t^{2}-\frac{9}{4}t=\frac{7}{16}$$

-\frac{9}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{9}{8} olish uchun. Keyin, -\frac{9}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$t^{2}-\frac{9}{4}t+\left(-\frac{9}{8}\right)^{2}=\frac{7}{16}+\left(-\frac{9}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{9}{8} kvadratini chiqarish.

$$t^{2}-\frac{9}{4}t+\frac{81}{64}=\frac{7}{16}+\frac{81}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{16} ni \frac{81}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$t^{2}-\frac{9}{4}t+\frac{81}{64}=\frac{109}{64}$$

t^{2}-\frac{9}{4}t+\frac{81}{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.

$$\left(t-\frac{9}{8}\right)^{2}=\frac{109}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(t-\frac{9}{8}\right)^{2}}=\sqrt{\frac{109}{64}}$$

Qisqartirish.

$$t-\frac{9}{8}=\frac{\sqrt{109}}{8} t-\frac{9}{8}=-\frac{\sqrt{109}}{8}$$

\frac{9}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$t=\frac{\sqrt{109}+9}{8} t=\frac{9-\sqrt{109}}{8}$$