3x^2+2x+15=9 eching

$$3 { x }^{ 2 } +2x+15=9$$

x uchun yechish (complex solution)

$$x=\frac{-1+\sqrt{17}i}{3}\approx -0,333333333+1,374368542i$$

$$x=\frac{-\sqrt{17}i-1}{3}\approx -0,333333333-1,374368542i$$

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.

$$3x^{2}+2x+15=9$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$3x^{2}+2x+15-9=9-9$$

O‘zidan 9 ayirilsa 0 qoladi.

$$3x^{2}+2x+15-9=0$$

15 dan 9 ni ayirish.

$$3x^{2}+2x+6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, 2 ni b va 6 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 3\times 6}}{2\times 3}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 3\times 6}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-12\times 6}}{2\times 3}$$

-12 ni 6 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-72}}{2\times 3}$$

4 ni -72 ga qo'shish.

$$x=\frac{-2±\sqrt{-68}}{2\times 3}$$

-68 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{17}i}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{17}i}{6}$$

x=\frac{-2±2\sqrt{17}i}{6} tenglamasini yeching, bunda ± musbat. -2 ni 2i\sqrt{17} ga qo'shish.

$$x=\frac{-2+2\sqrt{17}i}{6}$$

-2+2i\sqrt{17} ni 6 ga bo'lish.

$$x=\frac{-1+\sqrt{17}i}{3}$$

x=\frac{-2±2\sqrt{17}i}{6} tenglamasini yeching, bunda ± manfiy. -2 dan 2i\sqrt{17} ni ayirish.

$$x=\frac{-2\sqrt{17}i-2}{6}$$

-2-2i\sqrt{17} ni 6 ga bo'lish.

$$x=\frac{-\sqrt{17}i-1}{3}$$

Tenglama yechildi.

$$x=\frac{-1+\sqrt{17}i}{3} x=\frac{-\sqrt{17}i-1}{3}$$

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.

$$3x^{2}+2x+15=9$$

Tenglamaning ikkala tarafidan 15 ni ayirish.

$$3x^{2}+2x+15-15=9-15$$

O‘zidan 15 ayirilsa 0 qoladi.

$$3x^{2}+2x=9-15$$

9 dan 15 ni ayirish.

$$3x^{2}+2x=-6$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}+2x}{3}=-\frac{6}{3}$$

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

$$x^{2}+\frac{2}{3}x=-\frac{6}{3}$$

-6 ni 3 ga bo'lish.

$$x^{2}+\frac{2}{3}x=-2$$

\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{3} olish uchun. Keyin, \frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{2}{3}x+\left(\frac{1}{3}\right)^{2}=-2+\left(\frac{1}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{3} kvadratini chiqarish.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=-2+\frac{1}{9}$$

-2 ni \frac{1}{9} ga qo'shish.

$$x^{2}+\frac{2}{3}x+\frac{1}{9}=-\frac{17}{9}$$

x^{2}+\frac{2}{3}x+\frac{1}{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.

$$\left(x+\frac{1}{3}\right)^{2}=-\frac{17}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{3}\right)^{2}}=\sqrt{-\frac{17}{9}}$$

Qisqartirish.

$$x+\frac{1}{3}=\frac{\sqrt{17}i}{3} x+\frac{1}{3}=-\frac{\sqrt{17}i}{3}$$

Tenglamaning ikkala tarafidan \frac{1}{3} ni ayirish.

$$x=\frac{-1+\sqrt{17}i}{3} x=\frac{-\sqrt{17}i-1}{3}$$