2x^2-3/2x+7/10=0 eching

$$2 x ^ { 2 } - \frac { 3 } { 2 } x + \frac { 7 } { 10 } = 0$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8}\approx 0,375+0,45757513i$$

$$x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}\approx 0,375-0,45757513i$$

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.

$$2x^{2}-\frac{3}{2}x+\frac{7}{10}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 2 ni a, -\frac{3}{2} ni b va \frac{7}{10} ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\left(-\frac{3}{2}\right)^{2}-4\times 2\times \frac{7}{10}}}{2\times 2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{2} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-4\times 2\times \frac{7}{10}}}{2\times 2}$$

-4 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-8\times \frac{7}{10}}}{2\times 2}$$

-8 ni \frac{7}{10} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-\frac{28}{5}}}{2\times 2}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{9}{4} ni -\frac{28}{5} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{-\frac{67}{20}}}{2\times 2}$$

-\frac{67}{20} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\frac{\sqrt{335}i}{10}}{2\times 2}$$

-\frac{3}{2} ning teskarisi \frac{3}{2} ga teng.

$$x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{2\times 2}$$

2 ni 2 marotabaga ko'paytirish.

$$x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{4}$$

x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{4} tenglamasini yeching, bunda ± musbat. \frac{3}{2} ni \frac{i\sqrt{335}}{10} ga qo'shish.

$$x=\frac{\frac{\sqrt{335}i}{10}+\frac{3}{2}}{4}$$

\frac{3}{2}+\frac{i\sqrt{335}}{10} ni 4 ga bo'lish.

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8}$$

x=\frac{\frac{3}{2}±\frac{\sqrt{335}i}{10}}{4} tenglamasini yeching, bunda ± manfiy. \frac{3}{2} dan \frac{i\sqrt{335}}{10} ni ayirish.

$$x=\frac{-\frac{\sqrt{335}i}{10}+\frac{3}{2}}{4}$$

\frac{3}{2}-\frac{i\sqrt{335}}{10} ni 4 ga bo'lish.

$$x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}$$

Tenglama yechildi.

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8} x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}$$

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.

$$2x^{2}-\frac{3}{2}x+\frac{7}{10}=0$$

Tenglamaning ikkala tarafidan \frac{7}{10} ni ayirish.

$$2x^{2}-\frac{3}{2}x+\frac{7}{10}-\frac{7}{10}=-\frac{7}{10}$$

O‘zidan \frac{7}{10} ayirilsa 0 qoladi.

$$2x^{2}-\frac{3}{2}x=-\frac{7}{10}$$

Ikki tarafini 2 ga bo‘ling.

$$\frac{2x^{2}-\frac{3}{2}x}{2}=-\frac{\frac{7}{10}}{2}$$

2 ga bo'lish 2 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\frac{3}{2}}{2}\right)x=-\frac{\frac{7}{10}}{2}$$

-\frac{3}{2} ni 2 ga bo'lish.

$$x^{2}-\frac{3}{4}x=-\frac{\frac{7}{10}}{2}$$

-\frac{7}{10} ni 2 ga bo'lish.

$$x^{2}-\frac{3}{4}x=-\frac{7}{20}$$

-\frac{3}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{3}{8} olish uchun. Keyin, -\frac{3}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{3}{4}x+\left(-\frac{3}{8}\right)^{2}=-\frac{7}{20}+\left(-\frac{3}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{8} kvadratini chiqarish.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{7}{20}+\frac{9}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{7}{20} ni \frac{9}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{4}x+\frac{9}{64}=-\frac{67}{320}$$

x^{2}-\frac{3}{4}x+\frac{9}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{8}\right)^{2}=-\frac{67}{320}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{8}\right)^{2}}=\sqrt{-\frac{67}{320}}$$

Qisqartirish.

$$x-\frac{3}{8}=\frac{\sqrt{335}i}{40} x-\frac{3}{8}=-\frac{\sqrt{335}i}{40}$$

\frac{3}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{335}i}{40}+\frac{3}{8} x=-\frac{\sqrt{335}i}{40}+\frac{3}{8}$$