3x^2+16x-5=0 eching

$$3 x ^ { 2 } + 16 x - 5 = 0$$

x uchun yechish

$$x=\frac{\sqrt{79}-8}{3}\approx 0,296064806$$

$$x=\frac{-\sqrt{79}-8}{3}\approx -5,629398139$$

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.

$$3x^{2}+16x-5=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 16 ni b va -5 ni c bilan almashtiring.

$$x=\frac{-16±\sqrt{16^{2}-4\times 3\left(-5\right)}}{2\times 3}$$

16 kvadratini chiqarish.

$$x=\frac{-16±\sqrt{256-4\times 3\left(-5\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256-12\left(-5\right)}}{2\times 3}$$

-12 ni -5 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256+60}}{2\times 3}$$

256 ni 60 ga qo'shish.

$$x=\frac{-16±\sqrt{316}}{2\times 3}$$

316 ning kvadrat ildizini chiqarish.

$$x=\frac{-16±2\sqrt{79}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-16±2\sqrt{79}}{6}$$

x=\frac{-16±2\sqrt{79}}{6} tenglamasini yeching, bunda ± musbat. -16 ni 2\sqrt{79} ga qo'shish.

$$x=\frac{2\sqrt{79}-16}{6}$$

-16+2\sqrt{79} ni 6 ga bo'lish.

$$x=\frac{\sqrt{79}-8}{3}$$

x=\frac{-16±2\sqrt{79}}{6} tenglamasini yeching, bunda ± manfiy. -16 dan 2\sqrt{79} ni ayirish.

$$x=\frac{-2\sqrt{79}-16}{6}$$

-16-2\sqrt{79} ni 6 ga bo'lish.

$$x=\frac{-\sqrt{79}-8}{3}$$

Tenglama yechildi.

$$x=\frac{\sqrt{79}-8}{3} x=\frac{-\sqrt{79}-8}{3}$$

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.

$$3x^{2}+16x-5=0$$

5 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}+16x-5-\left(-5\right)=-\left(-5\right)$$

O‘zidan -5 ayirilsa 0 qoladi.

$$3x^{2}+16x=-\left(-5\right)$$

0 dan -5 ni ayirish.

$$3x^{2}+16x=5$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+16x}{3}=\frac{5}{3}$$

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

$$x^{2}+\frac{16}{3}x=\frac{5}{3}$$

\frac{16}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{8}{3} olish uchun. Keyin, \frac{8}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{16}{3}x+\left(\frac{8}{3}\right)^{2}=\frac{5}{3}+\left(\frac{8}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{8}{3} kvadratini chiqarish.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{5}{3}+\frac{64}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{5}{3} ni \frac{64}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{16}{3}x+\frac{64}{9}=\frac{79}{9}$$

x^{2}+\frac{16}{3}x+\frac{64}{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.

$$\left(x+\frac{8}{3}\right)^{2}=\frac{79}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{8}{3}\right)^{2}}=\sqrt{\frac{79}{9}}$$

Qisqartirish.

$$x+\frac{8}{3}=\frac{\sqrt{79}}{3} x+\frac{8}{3}=-\frac{\sqrt{79}}{3}$$

Tenglamaning ikkala tarafidan \frac{8}{3} ni ayirish.

$$x=\frac{\sqrt{79}-8}{3} x=\frac{-\sqrt{79}-8}{3}$$