-9x^2-4x-6=0 eching

$$- 9 x ^ { 2 } - 4 x - 6 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-5\sqrt{2}i-2}{9}\approx -0,222222222-0,785674201i$$

$$x=\frac{-2+5\sqrt{2}i}{9}\approx -0,222222222+0,785674201i$$

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.

$$-9x^{2}-4x-6=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -9 ni a, -4 ni b va -6 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-9\right)\left(-6\right)}}{2\left(-9\right)}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-9\right)\left(-6\right)}}{2\left(-9\right)}$$

-4 ni -9 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+36\left(-6\right)}}{2\left(-9\right)}$$

36 ni -6 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16-216}}{2\left(-9\right)}$$

16 ni -216 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{-200}}{2\left(-9\right)}$$

-200 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±10\sqrt{2}i}{2\left(-9\right)}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±10\sqrt{2}i}{2\left(-9\right)}$$

2 ni -9 marotabaga ko'paytirish.

$$x=\frac{4±10\sqrt{2}i}{-18}$$

x=\frac{4±10\sqrt{2}i}{-18} tenglamasini yeching, bunda ± musbat. 4 ni 10i\sqrt{2} ga qo'shish.

$$x=\frac{4+10\sqrt{2}i}{-18}$$

4+10i\sqrt{2} ni -18 ga bo'lish.

$$x=\frac{-5\sqrt{2}i-2}{9}$$

x=\frac{4±10\sqrt{2}i}{-18} tenglamasini yeching, bunda ± manfiy. 4 dan 10i\sqrt{2} ni ayirish.

$$x=\frac{-10\sqrt{2}i+4}{-18}$$

4-10i\sqrt{2} ni -18 ga bo'lish.

$$x=\frac{-2+5\sqrt{2}i}{9}$$

Tenglama yechildi.

$$x=\frac{-5\sqrt{2}i-2}{9} x=\frac{-2+5\sqrt{2}i}{9}$$

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.

$$-9x^{2}-4x-6=0$$

6 ni tenglamaning ikkala tarafiga qo'shish.

$$-9x^{2}-4x-6-\left(-6\right)=-\left(-6\right)$$

O‘zidan -6 ayirilsa 0 qoladi.

$$-9x^{2}-4x=-\left(-6\right)$$

0 dan -6 ni ayirish.

$$-9x^{2}-4x=6$$

Ikki tarafini -9 ga bo‘ling.

$$\frac{-9x^{2}-4x}{-9}=\frac{6}{-9}$$

-9 ga bo'lish -9 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{4}{-9}\right)x=\frac{6}{-9}$$

-4 ni -9 ga bo'lish.

$$x^{2}+\frac{4}{9}x=\frac{6}{-9}$$

\frac{6}{-9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

\frac{4}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{2}{9} olish uchun. Keyin, \frac{2}{9} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{2}{9} kvadratini chiqarish.

$$x^{2}+\frac{4}{9}x+\frac{4}{81}=-\frac{2}{3}+\frac{4}{81}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2}{3} ni \frac{4}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{4}{9}x+\frac{4}{81}=-\frac{50}{81}$$

x^{2}+\frac{4}{9}x+\frac{4}{81} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{2}{9}\right)^{2}=-\frac{50}{81}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{2}{9}\right)^{2}}=\sqrt{-\frac{50}{81}}$$

Qisqartirish.

$$x+\frac{2}{9}=\frac{5\sqrt{2}i}{9} x+\frac{2}{9}=-\frac{5\sqrt{2}i}{9}$$

Tenglamaning ikkala tarafidan \frac{2}{9} ni ayirish.

$$x=\frac{-2+5\sqrt{2}i}{9} x=\frac{-5\sqrt{2}i-2}{9}$$