3b^2-8b-15=0 eching

$$3 b ^ { 2 } - 8 b - 15 = 0$$

b uchun yechish

$$b = \frac{\sqrt{61} + 4}{3} \approx 3,936749892$$

$$b=\frac{4-\sqrt{61}}{3}\approx -1,270083225$$

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.

$$3b^{2}-8b-15=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -8 ni b va -15 ni c bilan almashtiring.

$$b=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 3\left(-15\right)}}{2\times 3}$$

-8 kvadratini chiqarish.

$$b=\frac{-\left(-8\right)±\sqrt{64-4\times 3\left(-15\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$b=\frac{-\left(-8\right)±\sqrt{64-12\left(-15\right)}}{2\times 3}$$

-12 ni -15 marotabaga ko'paytirish.

$$b=\frac{-\left(-8\right)±\sqrt{64+180}}{2\times 3}$$

64 ni 180 ga qo'shish.

$$b=\frac{-\left(-8\right)±\sqrt{244}}{2\times 3}$$

244 ning kvadrat ildizini chiqarish.

$$b=\frac{-\left(-8\right)±2\sqrt{61}}{2\times 3}$$

-8 ning teskarisi 8 ga teng.

$$b=\frac{8±2\sqrt{61}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$b=\frac{8±2\sqrt{61}}{6}$$

b=\frac{8±2\sqrt{61}}{6} tenglamasini yeching, bunda ± musbat. 8 ni 2\sqrt{61} ga qo'shish.

$$b=\frac{2\sqrt{61}+8}{6}$$

8+2\sqrt{61} ni 6 ga bo'lish.

$$b=\frac{\sqrt{61}+4}{3}$$

b=\frac{8±2\sqrt{61}}{6} tenglamasini yeching, bunda ± manfiy. 8 dan 2\sqrt{61} ni ayirish.

$$b=\frac{8-2\sqrt{61}}{6}$$

8-2\sqrt{61} ni 6 ga bo'lish.

$$b=\frac{4-\sqrt{61}}{3}$$

Tenglama yechildi.

$$b=\frac{\sqrt{61}+4}{3} b=\frac{4-\sqrt{61}}{3}$$

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.

$$3b^{2}-8b-15=0$$

15 ni tenglamaning ikkala tarafiga qo'shish.

$$3b^{2}-8b-15-\left(-15\right)=-\left(-15\right)$$

O‘zidan -15 ayirilsa 0 qoladi.

$$3b^{2}-8b=-\left(-15\right)$$

0 dan -15 ni ayirish.

$$3b^{2}-8b=15$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3b^{2}-8b}{3}=\frac{15}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$b^{2}-\frac{8}{3}b=\frac{15}{3}$$

15 ni 3 ga bo'lish.

$$b^{2}-\frac{8}{3}b=5$$

-\frac{8}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{3} olish uchun. Keyin, -\frac{4}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$b^{2}-\frac{8}{3}b+\left(-\frac{4}{3}\right)^{2}=5+\left(-\frac{4}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{3} kvadratini chiqarish.

$$b^{2}-\frac{8}{3}b+\frac{16}{9}=5+\frac{16}{9}$$

5 ni \frac{16}{9} ga qo'shish.

$$b^{2}-\frac{8}{3}b+\frac{16}{9}=\frac{61}{9}$$

b^{2}-\frac{8}{3}b+\frac{16}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(b-\frac{4}{3}\right)^{2}=\frac{61}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(b-\frac{4}{3}\right)^{2}}=\sqrt{\frac{61}{9}}$$

Qisqartirish.

$$b-\frac{4}{3}=\frac{\sqrt{61}}{3} b-\frac{4}{3}=-\frac{\sqrt{61}}{3}$$

\frac{4}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$b=\frac{\sqrt{61}+4}{3} b=\frac{4-\sqrt{61}}{3}$$