9x^2-48x+68=0 eching

$$9 x ^ { 2 } - 48 x + 68 = 0$$

x uchun yechish (complex solution)

$$x=\frac{8}{3}+\frac{2}{3}i\approx 2,666666667+0,666666667i$$

$$x=\frac{8}{3}-\frac{2}{3}i\approx 2,666666667-0,666666667i$$

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.

$$9x^{2}-48x+68=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 9 ni a, -48 ni b va 68 ni c bilan almashtiring.

$$x=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\times 9\times 68}}{2\times 9}$$

-48 kvadratini chiqarish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-4\times 9\times 68}}{2\times 9}$$

-4 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-36\times 68}}{2\times 9}$$

-36 ni 68 marotabaga ko'paytirish.

$$x=\frac{-\left(-48\right)±\sqrt{2304-2448}}{2\times 9}$$

2304 ni -2448 ga qo'shish.

$$x=\frac{-\left(-48\right)±\sqrt{-144}}{2\times 9}$$

-144 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-48\right)±12i}{2\times 9}$$

-48 ning teskarisi 48 ga teng.

$$x=\frac{48±12i}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$x=\frac{48±12i}{18}$$

x=\frac{48±12i}{18} tenglamasini yeching, bunda ± musbat. 48 ni 12i ga qo'shish.

$$x=\frac{48+12i}{18}$$

48+12i ni 18 ga bo'lish.

$$x=\frac{8}{3}+\frac{2}{3}i$$

x=\frac{48±12i}{18} tenglamasini yeching, bunda ± manfiy. 48 dan 12i ni ayirish.

$$x=\frac{48-12i}{18}$$

48-12i ni 18 ga bo'lish.

$$x=\frac{8}{3}-\frac{2}{3}i$$

Tenglama yechildi.

$$x=\frac{8}{3}+\frac{2}{3}i x=\frac{8}{3}-\frac{2}{3}i$$

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.

$$9x^{2}-48x+68=0$$

Tenglamaning ikkala tarafidan 68 ni ayirish.

$$9x^{2}-48x+68-68=-68$$

O‘zidan 68 ayirilsa 0 qoladi.

$$9x^{2}-48x=-68$$

Ikki tarafini 9 ga bo‘ling.

$$\frac{9x^{2}-48x}{9}=-\frac{68}{9}$$

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

$$x^{2}+\left(-\frac{48}{9}\right)x=-\frac{68}{9}$$

\frac{-48}{9} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{16}{3}x=-\frac{68}{9}$$

-\frac{16}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{8}{3} olish uchun. Keyin, -\frac{8}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{16}{3}x+\left(-\frac{8}{3}\right)^{2}=-\frac{68}{9}+\left(-\frac{8}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{8}{3} kvadratini chiqarish.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=\frac{-68+64}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{68}{9} ni \frac{64}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{16}{3}x+\frac{64}{9}=-\frac{4}{9}$$

x^{2}-\frac{16}{3}x+\frac{64}{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.

$$\left(x-\frac{8}{3}\right)^{2}=-\frac{4}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{8}{3}\right)^{2}}=\sqrt{-\frac{4}{9}}$$

Qisqartirish.

$$x-\frac{8}{3}=\frac{2}{3}i x-\frac{8}{3}=-\frac{2}{3}i$$

\frac{8}{3} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{8}{3}+\frac{2}{3}i x=\frac{8}{3}-\frac{2}{3}i$$