3x^2-4x-9=0 eching

$$3 { x }^{ 2 } -4x-9=0$$

x uchun yechish

$$x = \frac{\sqrt{31} + 2}{3} \approx 2,522588121$$

$$x=\frac{2-\sqrt{31}}{3}\approx -1,189254788$$

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.

$$3x^{2}-4x-9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -4 ni b va -9 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-9\right)}}{2\times 3}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-9\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-12\left(-9\right)}}{2\times 3}$$

-12 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+108}}{2\times 3}$$

16 ni 108 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{124}}{2\times 3}$$

124 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±2\sqrt{31}}{2\times 3}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±2\sqrt{31}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{4±2\sqrt{31}}{6}$$

x=\frac{4±2\sqrt{31}}{6} tenglamasini yeching, bunda ± musbat. 4 ni 2\sqrt{31} ga qo'shish.

$$x=\frac{2\sqrt{31}+4}{6}$$

4+2\sqrt{31} ni 6 ga bo'lish.

$$x=\frac{\sqrt{31}+2}{3}$$

x=\frac{4±2\sqrt{31}}{6} tenglamasini yeching, bunda ± manfiy. 4 dan 2\sqrt{31} ni ayirish.

$$x=\frac{4-2\sqrt{31}}{6}$$

4-2\sqrt{31} ni 6 ga bo'lish.

$$x=\frac{2-\sqrt{31}}{3}$$

Tenglama yechildi.

$$x=\frac{\sqrt{31}+2}{3} x=\frac{2-\sqrt{31}}{3}$$

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.

$$3x^{2}-4x-9=0$$

9 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}-4x-9-\left(-9\right)=-\left(-9\right)$$

O‘zidan -9 ayirilsa 0 qoladi.

$$3x^{2}-4x=-\left(-9\right)$$

0 dan -9 ni ayirish.

$$3x^{2}-4x=9$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-4x}{3}=\frac{9}{3}$$

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

$$x^{2}-\frac{4}{3}x=\frac{9}{3}$$

9 ni 3 ga bo'lish.

$$x^{2}-\frac{4}{3}x=3$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{2}{3} kvadratini chiqarish.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=3+\frac{4}{9}$$

3 ni \frac{4}{9} ga qo'shish.

$$x^{2}-\frac{4}{3}x+\frac{4}{9}=\frac{31}{9}$$

x^{2}-\frac{4}{3}x+\frac{4}{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.

$$\left(x-\frac{2}{3}\right)^{2}=\frac{31}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{2}{3}\right)^{2}}=\sqrt{\frac{31}{9}}$$

Qisqartirish.

$$x-\frac{2}{3}=\frac{\sqrt{31}}{3} x-\frac{2}{3}=-\frac{\sqrt{31}}{3}$$

\frac{2}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{31}+2}{3} x=\frac{2-\sqrt{31}}{3}$$