6x^2+5/3x-21=0 eching

$$6 x ^ { 2 } + \frac { 5 } { 3 } x - 21 = 0$$

x uchun yechish

$$x = \frac{\sqrt{4561} - 5}{36} \approx 1,737088223$$

$$x=\frac{-\sqrt{4561}-5}{36}\approx -2,014866001$$

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.

$$6x^{2}+\frac{5}{3}x-21=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, \frac{5}{3} ni b va -21 ni c bilan almashtiring.

$$x=\frac{-\frac{5}{3}±\sqrt{\left(\frac{5}{3}\right)^{2}-4\times 6\left(-21\right)}}{2\times 6}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{3} kvadratini chiqarish.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}-4\times 6\left(-21\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}-24\left(-21\right)}}{2\times 6}$$

-24 ni -21 marotabaga ko'paytirish.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{25}{9}+504}}{2\times 6}$$

\frac{25}{9} ni 504 ga qo'shish.

$$x=\frac{-\frac{5}{3}±\sqrt{\frac{4561}{9}}}{2\times 6}$$

\frac{4561}{9} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{12}$$

x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{12} tenglamasini yeching, bunda ± musbat. -\frac{5}{3} ni \frac{\sqrt{4561}}{3} ga qo'shish.

$$x=\frac{\sqrt{4561}-5}{3\times 12}$$

\frac{-5+\sqrt{4561}}{3} ni 12 ga bo'lish.

$$x=\frac{\sqrt{4561}-5}{36}$$

x=\frac{-\frac{5}{3}±\frac{\sqrt{4561}}{3}}{12} tenglamasini yeching, bunda ± manfiy. -\frac{5}{3} dan \frac{\sqrt{4561}}{3} ni ayirish.

$$x=\frac{-\sqrt{4561}-5}{3\times 12}$$

\frac{-5-\sqrt{4561}}{3} ni 12 ga bo'lish.

$$x=\frac{-\sqrt{4561}-5}{36}$$

Tenglama yechildi.

$$x=\frac{\sqrt{4561}-5}{36} x=\frac{-\sqrt{4561}-5}{36}$$

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.

$$6x^{2}+\frac{5}{3}x-21=0$$

21 ni tenglamaning ikkala tarafiga qo'shish.

$$6x^{2}+\frac{5}{3}x-21-\left(-21\right)=-\left(-21\right)$$

O‘zidan -21 ayirilsa 0 qoladi.

$$6x^{2}+\frac{5}{3}x=-\left(-21\right)$$

0 dan -21 ni ayirish.

$$6x^{2}+\frac{5}{3}x=21$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}+\frac{5}{3}x}{6}=\frac{21}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{\frac{5}{3}}{6}x=\frac{21}{6}$$

\frac{5}{3} ni 6 ga bo'lish.

$$x^{2}+\frac{5}{18}x=\frac{21}{6}$$

\frac{21}{6} ulushini 3 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{5}{18}x=\frac{7}{2}$$

\frac{5}{18} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{5}{36} olish uchun. Keyin, \frac{5}{36} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{5}{18}x+\left(\frac{5}{36}\right)^{2}=\frac{7}{2}+\left(\frac{5}{36}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{5}{36} kvadratini chiqarish.

$$x^{2}+\frac{5}{18}x+\frac{25}{1296}=\frac{7}{2}+\frac{25}{1296}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{7}{2} ni \frac{25}{1296} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{5}{18}x+\frac{25}{1296}=\frac{4561}{1296}$$

x^{2}+\frac{5}{18}x+\frac{25}{1296} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{5}{36}\right)^{2}=\frac{4561}{1296}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{5}{36}\right)^{2}}=\sqrt{\frac{4561}{1296}}$$

Qisqartirish.

$$x+\frac{5}{36}=\frac{\sqrt{4561}}{36} x+\frac{5}{36}=-\frac{\sqrt{4561}}{36}$$

Tenglamaning ikkala tarafidan \frac{5}{36} ni ayirish.

$$x=\frac{\sqrt{4561}-5}{36} x=\frac{-\sqrt{4561}-5}{36}$$