(1/2-x)*x=[2/7(1-1/5)]:[(1-3/5):(1+frac{2}{5})] eching

$$( \frac { 1 } { 2 } - x ) \cdot x = [ \frac { 2 } { 7 } ( 1 - \frac { 1 } { 5 } ) ] : [ ( 1 - \frac { 3 } { 5 } ) : ( 1 + \frac { 2 } { 5 } ) ]$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{295}i}{20}+\frac{1}{4}\approx 0,25+0,858778202i$$

$$x=-\frac{\sqrt{295}i}{20}+\frac{1}{4}\approx 0,25-0,858778202i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{1}{2}-x ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(1-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

1 ni \frac{5}{5} kasrga o‘giring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(\frac{5}{5}-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

\frac{5}{5} va \frac{1}{5} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{5-1}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

4 olish uchun 5 dan 1 ni ayirish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{4}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{2}{7} ni \frac{4}{5} ga ko‘paytiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2\times 4}{7\times 5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

\frac{2\times 4}{7\times 5} kasridagi ko‘paytirishlarni bajaring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

1 ni \frac{5}{5} kasrga o‘giring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5}{5}-\frac{3}{5}}{1+\frac{2}{5}}}$$

\frac{5}{5} va \frac{3}{5} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5-3}{5}}{1+\frac{2}{5}}}$$

2 olish uchun 5 dan 3 ni ayirish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{1+\frac{2}{5}}}$$

1 ni \frac{5}{5} kasrga o‘giring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5}{5}+\frac{2}{5}}}$$

\frac{5}{5} va \frac{2}{5} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5+2}{5}}}$$

7 olish uchun 5 va 2'ni qo'shing.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{7}{5}}}$$

\frac{2}{5} ni \frac{7}{5} ga bo'lish \frac{2}{5} ga k'paytirish \frac{7}{5} ga qaytarish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{5}\times \frac{5}{7}}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{2}{5} ni \frac{5}{7} ga ko‘paytiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2\times 5}{5\times 7}}$$

Surat va maxrajdagi ikkala 5 ni qisqartiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{7}}$$

\frac{8}{35} ni \frac{2}{7} ga bo'lish \frac{8}{35} ga k'paytirish \frac{2}{7} ga qaytarish.

$$\frac{1}{2}x-x^{2}=\frac{8}{35}\times \frac{7}{2}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{8}{35} ni \frac{7}{2} ga ko‘paytiring.

$$\frac{1}{2}x-x^{2}=\frac{8\times 7}{35\times 2}$$

\frac{8\times 7}{35\times 2} kasridagi ko‘paytirishlarni bajaring.

$$\frac{1}{2}x-x^{2}=\frac{56}{70}$$

\frac{56}{70} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{1}{2}x-x^{2}=\frac{4}{5}$$

Ikkala tarafdan \frac{4}{5} ni ayirish.

$$\frac{1}{2}x-x^{2}-\frac{4}{5}=0$$

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.

$$-x^{2}+\frac{1}{2}x-\frac{4}{5}=0$$

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

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

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{2} kvadratini chiqarish.

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

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}+4\left(-\frac{4}{5}\right)}}{2\left(-1\right)}$$

4 ni -\frac{4}{5} marotabaga ko'paytirish.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-\frac{16}{5}}}{2\left(-1\right)}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{4} ni -\frac{16}{5} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\frac{1}{2}±\sqrt{-\frac{59}{20}}}{2\left(-1\right)}$$

-\frac{59}{20} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{-2}$$

x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{-2} tenglamasini yeching, bunda ± musbat. -\frac{1}{2} ni \frac{i\sqrt{295}}{10} ga qo'shish.

$$x=\frac{\frac{\sqrt{295}i}{10}-\frac{1}{2}}{-2}$$

-\frac{1}{2}+\frac{i\sqrt{295}}{10} ni -2 ga bo'lish.

$$x=-\frac{\sqrt{295}i}{20}+\frac{1}{4}$$

x=\frac{-\frac{1}{2}±\frac{\sqrt{295}i}{10}}{-2} tenglamasini yeching, bunda ± manfiy. -\frac{1}{2} dan \frac{i\sqrt{295}}{10} ni ayirish.

$$x=\frac{-\frac{\sqrt{295}i}{10}-\frac{1}{2}}{-2}$$

-\frac{1}{2}-\frac{i\sqrt{295}}{10} ni -2 ga bo'lish.

$$x=\frac{\sqrt{295}i}{20}+\frac{1}{4}$$

Tenglama yechildi.

$$x=-\frac{\sqrt{295}i}{20}+\frac{1}{4} x=\frac{\sqrt{295}i}{20}+\frac{1}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{1}{2}-x ga x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(1-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

1 ni \frac{5}{5} kasrga o‘giring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\left(\frac{5}{5}-\frac{1}{5}\right)}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

\frac{5}{5} va \frac{1}{5} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{5-1}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

4 olish uchun 5 dan 1 ni ayirish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2}{7}\times \frac{4}{5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{2}{7} ni \frac{4}{5} ga ko‘paytiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{2\times 4}{7\times 5}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

\frac{2\times 4}{7\times 5} kasridagi ko‘paytirishlarni bajaring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{1-\frac{3}{5}}{1+\frac{2}{5}}}$$

1 ni \frac{5}{5} kasrga o‘giring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5}{5}-\frac{3}{5}}{1+\frac{2}{5}}}$$

\frac{5}{5} va \frac{3}{5} da bir xil maxraji bor, ularning suratini ayirish orqali ayiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{5-3}{5}}{1+\frac{2}{5}}}$$

2 olish uchun 5 dan 3 ni ayirish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{1+\frac{2}{5}}}$$

1 ni \frac{5}{5} kasrga o‘giring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5}{5}+\frac{2}{5}}}$$

\frac{5}{5} va \frac{2}{5} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{5+2}{5}}}$$

7 olish uchun 5 va 2'ni qo'shing.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{\frac{2}{5}}{\frac{7}{5}}}$$

\frac{2}{5} ni \frac{7}{5} ga bo'lish \frac{2}{5} ga k'paytirish \frac{7}{5} ga qaytarish.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{5}\times \frac{5}{7}}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{2}{5} ni \frac{5}{7} ga ko‘paytiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2\times 5}{5\times 7}}$$

Surat va maxrajdagi ikkala 5 ni qisqartiring.

$$\frac{1}{2}x-x^{2}=\frac{\frac{8}{35}}{\frac{2}{7}}$$

\frac{8}{35} ni \frac{2}{7} ga bo'lish \frac{8}{35} ga k'paytirish \frac{2}{7} ga qaytarish.

$$\frac{1}{2}x-x^{2}=\frac{8}{35}\times \frac{7}{2}$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{8}{35} ni \frac{7}{2} ga ko‘paytiring.

$$\frac{1}{2}x-x^{2}=\frac{8\times 7}{35\times 2}$$

\frac{8\times 7}{35\times 2} kasridagi ko‘paytirishlarni bajaring.

$$\frac{1}{2}x-x^{2}=\frac{56}{70}$$

\frac{56}{70} ulushini 14 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{1}{2}x-x^{2}=\frac{4}{5}$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-x^{2}+\frac{1}{2}x=\frac{4}{5}$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+\frac{1}{2}x}{-1}=\frac{\frac{4}{5}}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{\frac{1}{2}}{-1}x=\frac{\frac{4}{5}}{-1}$$

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

$$x^{2}-\frac{1}{2}x=\frac{\frac{4}{5}}{-1}$$

\frac{4}{5} ni -1 ga bo'lish.

$$x^{2}-\frac{1}{2}x=-\frac{4}{5}$$

-\frac{1}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{4} olish uchun. Keyin, -\frac{1}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=-\frac{4}{5}+\left(-\frac{1}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{4} kvadratini chiqarish.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{4}{5}+\frac{1}{16}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{4}{5} ni \frac{1}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=-\frac{59}{80}$$

x^{2}-\frac{1}{2}x+\frac{1}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{4}\right)^{2}=-\frac{59}{80}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{-\frac{59}{80}}$$

Qisqartirish.

$$x-\frac{1}{4}=\frac{\sqrt{295}i}{20} x-\frac{1}{4}=-\frac{\sqrt{295}i}{20}$$

\frac{1}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{295}i}{20}+\frac{1}{4} x=-\frac{\sqrt{295}i}{20}+\frac{1}{4}$$