27x^2+33x-120=0 eching

$$27 { x }^{ 2 } +33x-120=0$$

x uchun yechish

$$x = \frac{\sqrt{1561} - 11}{18} \approx 1,583860696$$

$$x=\frac{-\sqrt{1561}-11}{18}\approx -2,806082918$$

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.

$$27x^{2}+33x-120=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 27 ni a, 33 ni b va -120 ni c bilan almashtiring.

$$x=\frac{-33±\sqrt{33^{2}-4\times 27\left(-120\right)}}{2\times 27}$$

33 kvadratini chiqarish.

$$x=\frac{-33±\sqrt{1089-4\times 27\left(-120\right)}}{2\times 27}$$

-4 ni 27 marotabaga ko'paytirish.

$$x=\frac{-33±\sqrt{1089-108\left(-120\right)}}{2\times 27}$$

-108 ni -120 marotabaga ko'paytirish.

$$x=\frac{-33±\sqrt{1089+12960}}{2\times 27}$$

1089 ni 12960 ga qo'shish.

$$x=\frac{-33±\sqrt{14049}}{2\times 27}$$

14049 ning kvadrat ildizini chiqarish.

$$x=\frac{-33±3\sqrt{1561}}{2\times 27}$$

2 ni 27 marotabaga ko'paytirish.

$$x=\frac{-33±3\sqrt{1561}}{54}$$

x=\frac{-33±3\sqrt{1561}}{54} tenglamasini yeching, bunda ± musbat. -33 ni 3\sqrt{1561} ga qo'shish.

$$x=\frac{3\sqrt{1561}-33}{54}$$

-33+3\sqrt{1561} ni 54 ga bo'lish.

$$x=\frac{\sqrt{1561}-11}{18}$$

x=\frac{-33±3\sqrt{1561}}{54} tenglamasini yeching, bunda ± manfiy. -33 dan 3\sqrt{1561} ni ayirish.

$$x=\frac{-3\sqrt{1561}-33}{54}$$

-33-3\sqrt{1561} ni 54 ga bo'lish.

$$x=\frac{-\sqrt{1561}-11}{18}$$

Tenglama yechildi.

$$x=\frac{\sqrt{1561}-11}{18} x=\frac{-\sqrt{1561}-11}{18}$$

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.

$$27x^{2}+33x-120=0$$

120 ni tenglamaning ikkala tarafiga qo'shish.

$$27x^{2}+33x-120-\left(-120\right)=-\left(-120\right)$$

O‘zidan -120 ayirilsa 0 qoladi.

$$27x^{2}+33x=-\left(-120\right)$$

0 dan -120 ni ayirish.

$$27x^{2}+33x=120$$

Ikki tarafini 27 ga bo‘ling.

$$\frac{27x^{2}+33x}{27}=\frac{120}{27}$$

27 ga bo'lish 27 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{33}{27}x=\frac{120}{27}$$

\frac{33}{27} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{11}{9}x=\frac{120}{27}$$

\frac{120}{27} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{11}{9}x=\frac{40}{9}$$

\frac{11}{9} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{11}{18} olish uchun. Keyin, \frac{11}{18} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{11}{9}x+\left(\frac{11}{18}\right)^{2}=\frac{40}{9}+\left(\frac{11}{18}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{11}{18} kvadratini chiqarish.

$$x^{2}+\frac{11}{9}x+\frac{121}{324}=\frac{40}{9}+\frac{121}{324}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{40}{9} ni \frac{121}{324} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{11}{9}x+\frac{121}{324}=\frac{1561}{324}$$

x^{2}+\frac{11}{9}x+\frac{121}{324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{11}{18}\right)^{2}=\frac{1561}{324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{11}{18}\right)^{2}}=\sqrt{\frac{1561}{324}}$$

Qisqartirish.

$$x+\frac{11}{18}=\frac{\sqrt{1561}}{18} x+\frac{11}{18}=-\frac{\sqrt{1561}}{18}$$

Tenglamaning ikkala tarafidan \frac{11}{18} ni ayirish.

$$x=\frac{\sqrt{1561}-11}{18} x=\frac{-\sqrt{1561}-11}{18}$$