3x^2-14x+9=0 eching

$$3 x ^ { 2 } - 14 x + 9 = 0$$

x uchun yechish

$$x = \frac{\sqrt{22} + 7}{3} \approx 3,896805253$$

$$x=\frac{7-\sqrt{22}}{3}\approx 0,769861413$$

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.

$$3x^{2}-14x+9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -14 ni b va 9 ni c bilan almashtiring.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 3\times 9}}{2\times 3}$$

-14 kvadratini chiqarish.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 3\times 9}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-12\times 9}}{2\times 3}$$

-12 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-14\right)±\sqrt{196-108}}{2\times 3}$$

196 ni -108 ga qo'shish.

$$x=\frac{-\left(-14\right)±\sqrt{88}}{2\times 3}$$

88 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-14\right)±2\sqrt{22}}{2\times 3}$$

-14 ning teskarisi 14 ga teng.

$$x=\frac{14±2\sqrt{22}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{14±2\sqrt{22}}{6}$$

x=\frac{14±2\sqrt{22}}{6} tenglamasini yeching, bunda ± musbat. 14 ni 2\sqrt{22} ga qo'shish.

$$x=\frac{2\sqrt{22}+14}{6}$$

14+2\sqrt{22} ni 6 ga bo'lish.

$$x=\frac{\sqrt{22}+7}{3}$$

x=\frac{14±2\sqrt{22}}{6} tenglamasini yeching, bunda ± manfiy. 14 dan 2\sqrt{22} ni ayirish.

$$x=\frac{14-2\sqrt{22}}{6}$$

14-2\sqrt{22} ni 6 ga bo'lish.

$$x=\frac{7-\sqrt{22}}{3}$$

Tenglama yechildi.

$$x=\frac{\sqrt{22}+7}{3} x=\frac{7-\sqrt{22}}{3}$$

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.

$$3x^{2}-14x+9=0$$

Tenglamaning ikkala tarafidan 9 ni ayirish.

$$3x^{2}-14x+9-9=-9$$

O‘zidan 9 ayirilsa 0 qoladi.

$$3x^{2}-14x=-9$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-14x}{3}=-\frac{9}{3}$$

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

$$x^{2}-\frac{14}{3}x=-\frac{9}{3}$$

-9 ni 3 ga bo'lish.

$$x^{2}-\frac{14}{3}x=-3$$

-\frac{14}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{3} olish uchun. Keyin, -\frac{7}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{14}{3}x+\left(-\frac{7}{3}\right)^{2}=-3+\left(-\frac{7}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{3} kvadratini chiqarish.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=-3+\frac{49}{9}$$

-3 ni \frac{49}{9} ga qo'shish.

$$x^{2}-\frac{14}{3}x+\frac{49}{9}=\frac{22}{9}$$

x^{2}-\frac{14}{3}x+\frac{49}{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.

$$\left(x-\frac{7}{3}\right)^{2}=\frac{22}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{3}\right)^{2}}=\sqrt{\frac{22}{9}}$$

Qisqartirish.

$$x-\frac{7}{3}=\frac{\sqrt{22}}{3} x-\frac{7}{3}=-\frac{\sqrt{22}}{3}$$

\frac{7}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{22}+7}{3} x=\frac{7-\sqrt{22}}{3}$$