-5z^2+z+12=0 eching

$$- 5 z ^ { 2 } + z + 12 = 0$$

z uchun yechish

$$z = \frac{\sqrt{241} + 1}{10} \approx 1,65241747$$

$$z=\frac{1-\sqrt{241}}{10}\approx -1,45241747$$

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.

$$-5z^{2}+z+12=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, 1 ni b va 12 ni c bilan almashtiring.

$$z=\frac{-1±\sqrt{1^{2}-4\left(-5\right)\times 12}}{2\left(-5\right)}$$

1 kvadratini chiqarish.

$$z=\frac{-1±\sqrt{1-4\left(-5\right)\times 12}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$z=\frac{-1±\sqrt{1+20\times 12}}{2\left(-5\right)}$$

20 ni 12 marotabaga ko'paytirish.

$$z=\frac{-1±\sqrt{1+240}}{2\left(-5\right)}$$

1 ni 240 ga qo'shish.

$$z=\frac{-1±\sqrt{241}}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$z=\frac{-1±\sqrt{241}}{-10}$$

z=\frac{-1±\sqrt{241}}{-10} tenglamasini yeching, bunda ± musbat. -1 ni \sqrt{241} ga qo'shish.

$$z=\frac{\sqrt{241}-1}{-10}$$

-1+\sqrt{241} ni -10 ga bo'lish.

$$z=\frac{1-\sqrt{241}}{10}$$

z=\frac{-1±\sqrt{241}}{-10} tenglamasini yeching, bunda ± manfiy. -1 dan \sqrt{241} ni ayirish.

$$z=\frac{-\sqrt{241}-1}{-10}$$

-1-\sqrt{241} ni -10 ga bo'lish.

$$z=\frac{\sqrt{241}+1}{10}$$

Tenglama yechildi.

$$z=\frac{1-\sqrt{241}}{10} z=\frac{\sqrt{241}+1}{10}$$

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.

$$-5z^{2}+z+12=0$$

Tenglamaning ikkala tarafidan 12 ni ayirish.

$$-5z^{2}+z+12-12=-12$$

O‘zidan 12 ayirilsa 0 qoladi.

$$-5z^{2}+z=-12$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5z^{2}+z}{-5}=-\frac{12}{-5}$$

-5 ga bo'lish -5 ga ko'paytirishni bekor qiladi.

$$z^{2}+\frac{1}{-5}z=-\frac{12}{-5}$$

1 ni -5 ga bo'lish.

$$z^{2}-\frac{1}{5}z=-\frac{12}{-5}$$

-12 ni -5 ga bo'lish.

$$z^{2}-\frac{1}{5}z=\frac{12}{5}$$

-\frac{1}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{10} olish uchun. Keyin, -\frac{1}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$z^{2}-\frac{1}{5}z+\left(-\frac{1}{10}\right)^{2}=\frac{12}{5}+\left(-\frac{1}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{10} kvadratini chiqarish.

$$z^{2}-\frac{1}{5}z+\frac{1}{100}=\frac{12}{5}+\frac{1}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{12}{5} ni \frac{1}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$z^{2}-\frac{1}{5}z+\frac{1}{100}=\frac{241}{100}$$

z^{2}-\frac{1}{5}z+\frac{1}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(z-\frac{1}{10}\right)^{2}=\frac{241}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(z-\frac{1}{10}\right)^{2}}=\sqrt{\frac{241}{100}}$$

Qisqartirish.

$$z-\frac{1}{10}=\frac{\sqrt{241}}{10} z-\frac{1}{10}=-\frac{\sqrt{241}}{10}$$

\frac{1}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$z=\frac{\sqrt{241}+1}{10} z=\frac{1-\sqrt{241}}{10}$$