7k^2+18k-27=0 eching

$$7 k ^ { 2 } + 18 k - 27 = 0$$

k uchun yechish

$$k = \frac{3 \sqrt{30} - 9}{7} \approx 1,061668104$$

$$k=\frac{-3\sqrt{30}-9}{7}\approx -3,633096675$$

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.

$$7k^{2}+18k-27=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 7 ni a, 18 ni b va -27 ni c bilan almashtiring.

$$k=\frac{-18±\sqrt{18^{2}-4\times 7\left(-27\right)}}{2\times 7}$$

18 kvadratini chiqarish.

$$k=\frac{-18±\sqrt{324-4\times 7\left(-27\right)}}{2\times 7}$$

-4 ni 7 marotabaga ko'paytirish.

$$k=\frac{-18±\sqrt{324-28\left(-27\right)}}{2\times 7}$$

-28 ni -27 marotabaga ko'paytirish.

$$k=\frac{-18±\sqrt{324+756}}{2\times 7}$$

324 ni 756 ga qo'shish.

$$k=\frac{-18±\sqrt{1080}}{2\times 7}$$

1080 ning kvadrat ildizini chiqarish.

$$k=\frac{-18±6\sqrt{30}}{2\times 7}$$

2 ni 7 marotabaga ko'paytirish.

$$k=\frac{-18±6\sqrt{30}}{14}$$

k=\frac{-18±6\sqrt{30}}{14} tenglamasini yeching, bunda ± musbat. -18 ni 6\sqrt{30} ga qo'shish.

$$k=\frac{6\sqrt{30}-18}{14}$$

-18+6\sqrt{30} ni 14 ga bo'lish.

$$k=\frac{3\sqrt{30}-9}{7}$$

k=\frac{-18±6\sqrt{30}}{14} tenglamasini yeching, bunda ± manfiy. -18 dan 6\sqrt{30} ni ayirish.

$$k=\frac{-6\sqrt{30}-18}{14}$$

-18-6\sqrt{30} ni 14 ga bo'lish.

$$k=\frac{-3\sqrt{30}-9}{7}$$

Tenglama yechildi.

$$k=\frac{3\sqrt{30}-9}{7} k=\frac{-3\sqrt{30}-9}{7}$$

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.

$$7k^{2}+18k-27=0$$

27 ni tenglamaning ikkala tarafiga qo'shish.

$$7k^{2}+18k-27-\left(-27\right)=-\left(-27\right)$$

O‘zidan -27 ayirilsa 0 qoladi.

$$7k^{2}+18k=-\left(-27\right)$$

0 dan -27 ni ayirish.

$$7k^{2}+18k=27$$

Ikki tarafini 7 ga bo‘ling.

$$\frac{7k^{2}+18k}{7}=\frac{27}{7}$$

7 ga bo'lish 7 ga ko'paytirishni bekor qiladi.

$$k^{2}+\frac{18}{7}k=\frac{27}{7}$$

\frac{18}{7} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{9}{7} olish uchun. Keyin, \frac{9}{7} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$k^{2}+\frac{18}{7}k+\left(\frac{9}{7}\right)^{2}=\frac{27}{7}+\left(\frac{9}{7}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{7} kvadratini chiqarish.

$$k^{2}+\frac{18}{7}k+\frac{81}{49}=\frac{27}{7}+\frac{81}{49}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{27}{7} ni \frac{81}{49} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$k^{2}+\frac{18}{7}k+\frac{81}{49}=\frac{270}{49}$$

k^{2}+\frac{18}{7}k+\frac{81}{49} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(k+\frac{9}{7}\right)^{2}=\frac{270}{49}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(k+\frac{9}{7}\right)^{2}}=\sqrt{\frac{270}{49}}$$

Qisqartirish.

$$k+\frac{9}{7}=\frac{3\sqrt{30}}{7} k+\frac{9}{7}=-\frac{3\sqrt{30}}{7}$$

Tenglamaning ikkala tarafidan \frac{9}{7} ni ayirish.

$$k=\frac{3\sqrt{30}-9}{7} k=\frac{-3\sqrt{30}-9}{7}$$