3x^2-11x-6=0 eching

$$3 { x }^{ 2 } -11x-6 = 0$$

x uchun yechish

$$x = \frac{\sqrt{193} + 11}{6} \approx 4,148740665$$

$$x=\frac{11-\sqrt{193}}{6}\approx -0,482073998$$

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.

$$3x^{2}-11x-6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -11 ni b va -6 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 3\left(-6\right)}}{2\times 3}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 3\left(-6\right)}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-12\left(-6\right)}}{2\times 3}$$

-12 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121+72}}{2\times 3}$$

121 ni 72 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{193}}{2\times 3}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±\sqrt{193}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{11±\sqrt{193}}{6}$$

x=\frac{11±\sqrt{193}}{6} tenglamasini yeching, bunda ± musbat. 11 ni \sqrt{193} ga qo'shish.

$$x=\frac{\sqrt{193}+11}{6}$$

x=\frac{11±\sqrt{193}}{6} tenglamasini yeching, bunda ± manfiy. 11 dan \sqrt{193} ni ayirish.

$$x=\frac{11-\sqrt{193}}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{193}+11}{6} x=\frac{11-\sqrt{193}}{6}$$

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.

$$3x^{2}-11x-6=0$$

6 ni tenglamaning ikkala tarafiga qo'shish.

$$3x^{2}-11x-6-\left(-6\right)=-\left(-6\right)$$

O‘zidan -6 ayirilsa 0 qoladi.

$$3x^{2}-11x=-\left(-6\right)$$

0 dan -6 ni ayirish.

$$3x^{2}-11x=6$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-11x}{3}=\frac{6}{3}$$

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

$$x^{2}-\frac{11}{3}x=\frac{6}{3}$$

6 ni 3 ga bo'lish.

$$x^{2}-\frac{11}{3}x=2$$

-\frac{11}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{6} olish uchun. Keyin, -\frac{11}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{3}x+\left(-\frac{11}{6}\right)^{2}=2+\left(-\frac{11}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{6} kvadratini chiqarish.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=2+\frac{121}{36}$$

2 ni \frac{121}{36} ga qo'shish.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=\frac{193}{36}$$

x^{2}-\frac{11}{3}x+\frac{121}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{6}\right)^{2}=\frac{193}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{6}\right)^{2}}=\sqrt{\frac{193}{36}}$$

Qisqartirish.

$$x-\frac{11}{6}=\frac{\sqrt{193}}{6} x-\frac{11}{6}=-\frac{\sqrt{193}}{6}$$

\frac{11}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{193}+11}{6} x=\frac{11-\sqrt{193}}{6}$$