1/2x^2-3/2x+2/1=0 eching

$$\frac { 1 } { 2 } x ^ { 2 } - \frac { 3 } { 2 } x + \frac { 2 } { 1 } = 0$$

x uchun yechish (complex solution)

$$x=\frac{3+\sqrt{7}i}{2}\approx 1,5+1,322875656i$$

$$x=\frac{-\sqrt{7}i+3}{2}\approx 1,5-1,322875656i$$

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.

$$\frac{1}{2}x^{2}-\frac{3}{2}x+2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{2} ni a, -\frac{3}{2} ni b va 2 ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\left(-\frac{3}{2}\right)^{2}-4\times \frac{1}{2}\times 2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-4\times \frac{1}{2}\times 2}}{2\times \frac{1}{2}}$$

-4 ni \frac{1}{2} marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-2\times 2}}{2\times \frac{1}{2}}$$

-2 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{\frac{9}{4}-4}}{2\times \frac{1}{2}}$$

\frac{9}{4} ni -4 ga qo'shish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\sqrt{-\frac{7}{4}}}{2\times \frac{1}{2}}$$

-\frac{7}{4} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{3}{2}\right)±\frac{\sqrt{7}i}{2}}{2\times \frac{1}{2}}$$

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

$$x=\frac{\frac{3}{2}±\frac{\sqrt{7}i}{2}}{2\times \frac{1}{2}}$$

2 ni \frac{1}{2} marotabaga ko'paytirish.

$$x=\frac{\frac{3}{2}±\frac{\sqrt{7}i}{2}}{1}$$

x=\frac{\frac{3}{2}±\frac{\sqrt{7}i}{2}}{1} tenglamasini yeching, bunda ± musbat. \frac{3}{2} ni \frac{i\sqrt{7}}{2} ga qo'shish.

$$x=\frac{3+\sqrt{7}i}{2}$$

x=\frac{\frac{3}{2}±\frac{\sqrt{7}i}{2}}{1} tenglamasini yeching, bunda ± manfiy. \frac{3}{2} dan \frac{i\sqrt{7}}{2} ni ayirish.

$$x=\frac{-\sqrt{7}i+3}{2}$$

Tenglama yechildi.

$$x=\frac{3+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+3}{2}$$

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.

$$\frac{1}{2}x^{2}-\frac{3}{2}x+2=0$$

Tenglamaning ikkala tarafidan 2 ni ayirish.

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

O‘zidan 2 ayirilsa 0 qoladi.

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

Ikkala tarafini 2 ga ko‘paytiring.

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

\frac{1}{2} ga bo'lish \frac{1}{2} ga ko'paytirishni bekor qiladi.

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

-\frac{3}{2} ni \frac{1}{2} ga bo'lish -\frac{3}{2} ga k'paytirish \frac{1}{2} ga qaytarish.

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

-2 ni \frac{1}{2} ga bo'lish -2 ga k'paytirish \frac{1}{2} ga qaytarish.

$$x^{2}-3x=-4$$

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

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

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

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

-4 ni \frac{9}{4} ga qo'shish.

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

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

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

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

Qisqartirish.

$$x-\frac{3}{2}=\frac{\sqrt{7}i}{2} x-\frac{3}{2}=-\frac{\sqrt{7}i}{2}$$

\frac{3}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{3+\sqrt{7}i}{2} x=\frac{-\sqrt{7}i+3}{2}$$