18x^2-30x+11=0 eching

$$18 x ^ { 2 } - 30 x + 11 = 0$$

x uchun yechish

$$x = \frac{\sqrt{3} + 5}{6} \approx 1,122008468$$

$$x=\frac{5-\sqrt{3}}{6}\approx 0,544658199$$

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.

$$18x^{2}-30x+11=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 18 ni a, -30 ni b va 11 ni c bilan almashtiring.

$$x=\frac{-\left(-30\right)±\sqrt{\left(-30\right)^{2}-4\times 18\times 11}}{2\times 18}$$

-30 kvadratini chiqarish.

$$x=\frac{-\left(-30\right)±\sqrt{900-4\times 18\times 11}}{2\times 18}$$

-4 ni 18 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900-72\times 11}}{2\times 18}$$

-72 ni 11 marotabaga ko'paytirish.

$$x=\frac{-\left(-30\right)±\sqrt{900-792}}{2\times 18}$$

900 ni -792 ga qo'shish.

$$x=\frac{-\left(-30\right)±\sqrt{108}}{2\times 18}$$

108 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-30\right)±6\sqrt{3}}{2\times 18}$$

-30 ning teskarisi 30 ga teng.

$$x=\frac{30±6\sqrt{3}}{2\times 18}$$

2 ni 18 marotabaga ko'paytirish.

$$x=\frac{30±6\sqrt{3}}{36}$$

x=\frac{30±6\sqrt{3}}{36} tenglamasini yeching, bunda ± musbat. 30 ni 6\sqrt{3} ga qo'shish.

$$x=\frac{6\sqrt{3}+30}{36}$$

30+6\sqrt{3} ni 36 ga bo'lish.

$$x=\frac{\sqrt{3}+5}{6}$$

x=\frac{30±6\sqrt{3}}{36} tenglamasini yeching, bunda ± manfiy. 30 dan 6\sqrt{3} ni ayirish.

$$x=\frac{30-6\sqrt{3}}{36}$$

30-6\sqrt{3} ni 36 ga bo'lish.

$$x=\frac{5-\sqrt{3}}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{3}+5}{6} x=\frac{5-\sqrt{3}}{6}$$

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.

$$18x^{2}-30x+11=0$$

Tenglamaning ikkala tarafidan 11 ni ayirish.

$$18x^{2}-30x+11-11=-11$$

O‘zidan 11 ayirilsa 0 qoladi.

$$18x^{2}-30x=-11$$

Ikki tarafini 18 ga bo‘ling.

$$\frac{18x^{2}-30x}{18}=-\frac{11}{18}$$

18 ga bo'lish 18 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{30}{18}\right)x=-\frac{11}{18}$$

\frac{-30}{18} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{5}{3}x=-\frac{11}{18}$$

-\frac{5}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{6} olish uchun. Keyin, -\frac{5}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=-\frac{11}{18}+\left(-\frac{5}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{6} kvadratini chiqarish.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=-\frac{11}{18}+\frac{25}{36}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{11}{18} ni \frac{25}{36} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{1}{12}$$

x^{2}-\frac{5}{3}x+\frac{25}{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.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{1}{12}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{1}{12}}$$

Qisqartirish.

$$x-\frac{5}{6}=\frac{\sqrt{3}}{6} x-\frac{5}{6}=-\frac{\sqrt{3}}{6}$$

\frac{5}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{3}+5}{6} x=\frac{5-\sqrt{3}}{6}$$