20x^2-28x-1=0 eching

$$20 x ^ { 2 } - 28 x - 1 = 0$$

x uchun yechish

$$x = \frac{3 \sqrt{6} + 7}{10} \approx 1,434846923$$

$$x=\frac{7-3\sqrt{6}}{10}\approx -0,034846923$$

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.

$$20x^{2}-28x-1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 20 ni a, -28 ni b va -1 ni c bilan almashtiring.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 20\left(-1\right)}}{2\times 20}$$

-28 kvadratini chiqarish.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 20\left(-1\right)}}{2\times 20}$$

-4 ni 20 marotabaga ko'paytirish.

$$x=\frac{-\left(-28\right)±\sqrt{784-80\left(-1\right)}}{2\times 20}$$

-80 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-28\right)±\sqrt{784+80}}{2\times 20}$$

784 ni 80 ga qo'shish.

$$x=\frac{-\left(-28\right)±\sqrt{864}}{2\times 20}$$

864 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-28\right)±12\sqrt{6}}{2\times 20}$$

-28 ning teskarisi 28 ga teng.

$$x=\frac{28±12\sqrt{6}}{2\times 20}$$

2 ni 20 marotabaga ko'paytirish.

$$x=\frac{28±12\sqrt{6}}{40}$$

x=\frac{28±12\sqrt{6}}{40} tenglamasini yeching, bunda ± musbat. 28 ni 12\sqrt{6} ga qo'shish.

$$x=\frac{12\sqrt{6}+28}{40}$$

28+12\sqrt{6} ni 40 ga bo'lish.

$$x=\frac{3\sqrt{6}+7}{10}$$

x=\frac{28±12\sqrt{6}}{40} tenglamasini yeching, bunda ± manfiy. 28 dan 12\sqrt{6} ni ayirish.

$$x=\frac{28-12\sqrt{6}}{40}$$

28-12\sqrt{6} ni 40 ga bo'lish.

$$x=\frac{7-3\sqrt{6}}{10}$$

Tenglama yechildi.

$$x=\frac{3\sqrt{6}+7}{10} x=\frac{7-3\sqrt{6}}{10}$$

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.

$$20x^{2}-28x-1=0$$

1 ni tenglamaning ikkala tarafiga qo'shish.

$$20x^{2}-28x-1-\left(-1\right)=-\left(-1\right)$$

O‘zidan -1 ayirilsa 0 qoladi.

$$20x^{2}-28x=-\left(-1\right)$$

0 dan -1 ni ayirish.

$$20x^{2}-28x=1$$

Ikki tarafini 20 ga bo‘ling.

$$\frac{20x^{2}-28x}{20}=\frac{1}{20}$$

20 ga bo'lish 20 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{28}{20}\right)x=\frac{1}{20}$$

\frac{-28}{20} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{5}x=\frac{1}{20}$$

-\frac{7}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{10} olish uchun. Keyin, -\frac{7}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{5}x+\left(-\frac{7}{10}\right)^{2}=\frac{1}{20}+\left(-\frac{7}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{10} kvadratini chiqarish.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{1}{20}+\frac{49}{100}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{20} ni \frac{49}{100} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{7}{5}x+\frac{49}{100}=\frac{27}{50}$$

x^{2}-\frac{7}{5}x+\frac{49}{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.

$$\left(x-\frac{7}{10}\right)^{2}=\frac{27}{50}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{10}\right)^{2}}=\sqrt{\frac{27}{50}}$$

Qisqartirish.

$$x-\frac{7}{10}=\frac{3\sqrt{6}}{10} x-\frac{7}{10}=-\frac{3\sqrt{6}}{10}$$

\frac{7}{10} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3\sqrt{6}+7}{10} x=\frac{7-3\sqrt{6}}{10}$$