3x^2+8x-3=65 eching

$$3 { x }^{ 2 } +8x-3=65$$

x uchun yechish

$$x = \frac{2 \sqrt{55} - 4}{3} \approx 3,610798991$$

$$x=\frac{-2\sqrt{55}-4}{3}\approx -6,277465658$$

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.

$$3x^{2}+8x-3=65$$

Tenglamaning ikkala tarafidan 65 ni ayirish.

$$3x^{2}+8x-3-65=65-65$$

O‘zidan 65 ayirilsa 0 qoladi.

$$3x^{2}+8x-3-65=0$$

-3 dan 65 ni ayirish.

$$3x^{2}+8x-68=0$$

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 -68 ni c bilan almashtiring.

$$x=\frac{-8±\sqrt{8^{2}-4\times 3\left(-68\right)}}{2\times 3}$$

8 kvadratini chiqarish.

$$x=\frac{-8±\sqrt{64-4\times 3\left(-68\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64-12\left(-68\right)}}{2\times 3}$$

-12 ni -68 marotabaga ko'paytirish.

$$x=\frac{-8±\sqrt{64+816}}{2\times 3}$$

64 ni 816 ga qo'shish.

$$x=\frac{-8±\sqrt{880}}{2\times 3}$$

880 ning kvadrat ildizini chiqarish.

$$x=\frac{-8±4\sqrt{55}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-8±4\sqrt{55}}{6}$$

x=\frac{-8±4\sqrt{55}}{6} tenglamasini yeching, bunda ± musbat. -8 ni 4\sqrt{55} ga qo'shish.

$$x=\frac{4\sqrt{55}-8}{6}$$

-8+4\sqrt{55} ni 6 ga bo'lish.

$$x=\frac{2\sqrt{55}-4}{3}$$

x=\frac{-8±4\sqrt{55}}{6} tenglamasini yeching, bunda ± manfiy. -8 dan 4\sqrt{55} ni ayirish.

$$x=\frac{-4\sqrt{55}-8}{6}$$

-8-4\sqrt{55} ni 6 ga bo'lish.

$$x=\frac{-2\sqrt{55}-4}{3}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{55}-4}{3} x=\frac{-2\sqrt{55}-4}{3}$$

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.

$$3x^{2}+8x-3=65$$

3 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}+8x-3-\left(-3\right)=65-\left(-3\right)$$

O‘zidan -3 ayirilsa 0 qoladi.

$$3x^{2}+8x=65-\left(-3\right)$$

65 dan -3 ni ayirish.

$$3x^{2}+8x=68$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+8x}{3}=\frac{68}{3}$$

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

$$x^{2}+\frac{8}{3}x=\frac{68}{3}$$

\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.

$$x^{2}+\frac{8}{3}x+\left(\frac{4}{3}\right)^{2}=\frac{68}{3}+\left(\frac{4}{3}\right)^{2}$$

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

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{68}{3}+\frac{16}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{68}{3} ni \frac{16}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{8}{3}x+\frac{16}{9}=\frac{220}{9}$$

x^{2}+\frac{8}{3}x+\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.

$$\left(x+\frac{4}{3}\right)^{2}=\frac{220}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{4}{3}\right)^{2}}=\sqrt{\frac{220}{9}}$$

Qisqartirish.

$$x+\frac{4}{3}=\frac{2\sqrt{55}}{3} x+\frac{4}{3}=-\frac{2\sqrt{55}}{3}$$

Tenglamaning ikkala tarafidan \frac{4}{3} ni ayirish.

$$x=\frac{2\sqrt{55}-4}{3} x=\frac{-2\sqrt{55}-4}{3}$$