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

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

x uchun yechish

$$x = \frac{2 \sqrt{7} + 1}{3} \approx 2,097167541$$

$$x=\frac{1-2\sqrt{7}}{3}\approx -1,430500874$$

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.

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

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 -9 ni c bilan almashtiring.

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

-2 kvadratini chiqarish.

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

-4 ni 3 marotabaga ko'paytirish.

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

-12 ni -9 marotabaga ko'paytirish.

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

4 ni 108 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{112}}{2\times 3}$$

112 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±4\sqrt{7}}{2\times 3}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±4\sqrt{7}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{2±4\sqrt{7}}{6}$$

x=\frac{2±4\sqrt{7}}{6} tenglamasini yeching, bunda ± musbat. 2 ni 4\sqrt{7} ga qo'shish.

$$x=\frac{4\sqrt{7}+2}{6}$$

2+4\sqrt{7} ni 6 ga bo'lish.

$$x=\frac{2\sqrt{7}+1}{3}$$

x=\frac{2±4\sqrt{7}}{6} tenglamasini yeching, bunda ± manfiy. 2 dan 4\sqrt{7} ni ayirish.

$$x=\frac{2-4\sqrt{7}}{6}$$

2-4\sqrt{7} ni 6 ga bo'lish.

$$x=\frac{1-2\sqrt{7}}{3}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{7}+1}{3} x=\frac{1-2\sqrt{7}}{3}$$

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.

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

9 ni tenglamaning ikkala tarafiga qo'shish.

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

O‘zidan -9 ayirilsa 0 qoladi.

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

0 dan -9 ni ayirish.

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

Ikki tarafini 3 ga bo‘ling.

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

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

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

9 ni 3 ga bo'lish.

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

-\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.

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

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

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

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

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{28}{9}$$

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.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{28}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{28}{9}}$$

Qisqartirish.

$$x-\frac{1}{3}=\frac{2\sqrt{7}}{3} x-\frac{1}{3}=-\frac{2\sqrt{7}}{3}$$

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{2\sqrt{7}+1}{3} x=\frac{1-2\sqrt{7}}{3}$$