9x^2+14x+21=0 eching

$$9 x ^ { 2 } + 14 x + 21 = 0$$

x uchun yechish (complex solution)

$$x=\frac{-7+2\sqrt{35}i}{9}\approx -0,777777778+1,314684396i$$

$$x=\frac{-2\sqrt{35}i-7}{9}\approx -0,777777778-1,314684396i$$

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.

$$9x^{2}+14x+21=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, 14 ni b va 21 ni c bilan almashtiring.

$$x=\frac{-14±\sqrt{14^{2}-4\times 9\times 21}}{2\times 9}$$

14 kvadratini chiqarish.

$$x=\frac{-14±\sqrt{196-4\times 9\times 21}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196-36\times 21}}{2\times 9}$$

-36 ni 21 marotabaga ko'paytirish.

$$x=\frac{-14±\sqrt{196-756}}{2\times 9}$$

196 ni -756 ga qo'shish.

$$x=\frac{-14±\sqrt{-560}}{2\times 9}$$

-560 ning kvadrat ildizini chiqarish.

$$x=\frac{-14±4\sqrt{35}i}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{-14±4\sqrt{35}i}{18}$$

x=\frac{-14±4\sqrt{35}i}{18} tenglamasini yeching, bunda ± musbat. -14 ni 4i\sqrt{35} ga qo'shish.

$$x=\frac{-14+4\sqrt{35}i}{18}$$

-14+4i\sqrt{35} ni 18 ga bo'lish.

$$x=\frac{-7+2\sqrt{35}i}{9}$$

x=\frac{-14±4\sqrt{35}i}{18} tenglamasini yeching, bunda ± manfiy. -14 dan 4i\sqrt{35} ni ayirish.

$$x=\frac{-4\sqrt{35}i-14}{18}$$

-14-4i\sqrt{35} ni 18 ga bo'lish.

$$x=\frac{-2\sqrt{35}i-7}{9}$$

Tenglama yechildi.

$$x=\frac{-7+2\sqrt{35}i}{9} x=\frac{-2\sqrt{35}i-7}{9}$$

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.

$$9x^{2}+14x+21=0$$

Tenglamaning ikkala tarafidan 21 ni ayirish.

$$9x^{2}+14x+21-21=-21$$

O‘zidan 21 ayirilsa 0 qoladi.

$$9x^{2}+14x=-21$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}+14x}{9}=-\frac{21}{9}$$

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

$$x^{2}+\frac{14}{9}x=-\frac{21}{9}$$

\frac{-21}{9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{14}{9}x=-\frac{7}{3}$$

\frac{14}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{7}{9} olish uchun. Keyin, \frac{7}{9} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{14}{9}x+\left(\frac{7}{9}\right)^{2}=-\frac{7}{3}+\left(\frac{7}{9}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{7}{9} kvadratini chiqarish.

$$x^{2}+\frac{14}{9}x+\frac{49}{81}=-\frac{7}{3}+\frac{49}{81}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{3} ni \frac{49}{81} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{14}{9}x+\frac{49}{81}=-\frac{140}{81}$$

x^{2}+\frac{14}{9}x+\frac{49}{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.

$$\left(x+\frac{7}{9}\right)^{2}=-\frac{140}{81}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{7}{9}\right)^{2}}=\sqrt{-\frac{140}{81}}$$

Qisqartirish.

$$x+\frac{7}{9}=\frac{2\sqrt{35}i}{9} x+\frac{7}{9}=-\frac{2\sqrt{35}i}{9}$$

Tenglamaning ikkala tarafidan \frac{7}{9} ni ayirish.

$$x=\frac{-7+2\sqrt{35}i}{9} x=\frac{-2\sqrt{35}i-7}{9}$$