3k(2k-1)=2 eching

$$3 k ( 2 k - 1 ) = 2$$

k uchun yechish

$$k=\frac{\sqrt{57}}{12}+\frac{1}{4}\approx 0,87915287$$

$$k=-\frac{\sqrt{57}}{12}+\frac{1}{4}\approx -0,37915287$$

Kvadrat tenglamalaridan foydalanish qoidalari

3k ga 2k-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6k^{2}-3k=2$$

Ikkala tarafdan 2 ni ayirish.

$$6k^{2}-3k-2=0$$

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

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

-3 kvadratini chiqarish.

$$k=\frac{-\left(-3\right)±\sqrt{9-4\times 6\left(-2\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$k=\frac{-\left(-3\right)±\sqrt{9-24\left(-2\right)}}{2\times 6}$$

-24 ni -2 marotabaga ko'paytirish.

$$k=\frac{-\left(-3\right)±\sqrt{9+48}}{2\times 6}$$

9 ni 48 ga qo'shish.

$$k=\frac{-\left(-3\right)±\sqrt{57}}{2\times 6}$$

-3 ning teskarisi 3 ga teng.

$$k=\frac{3±\sqrt{57}}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$k=\frac{3±\sqrt{57}}{12}$$

k=\frac{3±\sqrt{57}}{12} tenglamasini yeching, bunda ± musbat. 3 ni \sqrt{57} ga qo'shish.

$$k=\frac{\sqrt{57}+3}{12}$$

3+\sqrt{57} ni 12 ga bo'lish.

$$k=\frac{\sqrt{57}}{12}+\frac{1}{4}$$

k=\frac{3±\sqrt{57}}{12} tenglamasini yeching, bunda ± manfiy. 3 dan \sqrt{57} ni ayirish.

$$k=\frac{3-\sqrt{57}}{12}$$

3-\sqrt{57} ni 12 ga bo'lish.

$$k=-\frac{\sqrt{57}}{12}+\frac{1}{4}$$

Tenglama yechildi.

$$k=\frac{\sqrt{57}}{12}+\frac{1}{4} k=-\frac{\sqrt{57}}{12}+\frac{1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

3k ga 2k-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$6k^{2}-3k=2$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6k^{2}-3k}{6}=\frac{2}{6}$$

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

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

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

$$k^{2}-\frac{1}{2}k=\frac{2}{6}$$

\frac{2}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$k^{2}-\frac{1}{2}k=\frac{1}{3}$$

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$k^{2}-\frac{1}{2}k+\left(-\frac{1}{4}\right)^{2}=\frac{1}{3}+\left(-\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$k^{2}-\frac{1}{2}k+\frac{1}{16}=\frac{1}{3}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{3} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$k^{2}-\frac{1}{2}k+\frac{1}{16}=\frac{19}{48}$$

k^{2}-\frac{1}{2}k+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(k-\frac{1}{4}\right)^{2}=\frac{19}{48}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k-\frac{1}{4}\right)^{2}}=\sqrt{\frac{19}{48}}$$

Qisqartirish.

$$k-\frac{1}{4}=\frac{\sqrt{57}}{12} k-\frac{1}{4}=-\frac{\sqrt{57}}{12}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$k=\frac{\sqrt{57}}{12}+\frac{1}{4} k=-\frac{\sqrt{57}}{12}+\frac{1}{4}$$