1/5x-3=5x(1/10x+1/10) eching

$$\frac { 1 } { 5 } x - 3 = 5 x ( \frac { 1 } { 10 } x + \frac { 1 } { 10 } )$$

x uchun yechish (complex solution)

$$x=\frac{-3+\sqrt{591}i}{10}\approx -0,3+2,431049156i$$

$$x=\frac{-\sqrt{591}i-3}{10}\approx -0,3-2,431049156i$$

Kvadrat tenglamalaridan foydalanish qoidalari

5x ga \frac{1}{10}x+\frac{1}{10} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{1}{5}x-3=5x\times \frac{1}{10}x+5x\times \frac{1}{10}$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$\frac{1}{5}x-3=5x^{2}\times \frac{1}{10}+5x\times \frac{1}{10}$$

\frac{5}{10} hosil qilish uchun 5 va \frac{1}{10} ni ko'paytirish.

$$\frac{1}{5}x-3=\frac{5}{10}x^{2}+5x\times \frac{1}{10}$$

\frac{5}{10} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{1}{5}x-3=\frac{1}{2}x^{2}+5x\times \frac{1}{10}$$

\frac{5}{10} hosil qilish uchun 5 va \frac{1}{10} ni ko'paytirish.

$$\frac{1}{5}x-3=\frac{1}{2}x^{2}+\frac{5}{10}x$$

\frac{5}{10} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Ikkala tarafdan \frac{1}{2}x^{2} ni ayirish.

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

Ikkala tarafdan \frac{1}{2}x ni ayirish.

$$\frac{1}{5}x-3-\frac{1}{2}x^{2}-\frac{1}{2}x=0$$

-\frac{3}{10}x ni olish uchun \frac{1}{5}x va -\frac{1}{2}x ni birlashtirish.

$$-\frac{3}{10}x-3-\frac{1}{2}x^{2}=0$$

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.

$$-\frac{1}{2}x^{2}-\frac{3}{10}x-3=0$$

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}{10} ni b va -3 ni c bilan almashtiring.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\left(-\frac{3}{10}\right)^{2}-4\left(-\frac{1}{2}\right)\left(-3\right)}}{2\left(-\frac{1}{2}\right)}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{10} kvadratini chiqarish.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-4\left(-\frac{1}{2}\right)\left(-3\right)}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}+2\left(-3\right)}}{2\left(-\frac{1}{2}\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{\frac{9}{100}-6}}{2\left(-\frac{1}{2}\right)}$$

\frac{9}{100} ni -6 ga qo'shish.

$$x=\frac{-\left(-\frac{3}{10}\right)±\sqrt{-\frac{591}{100}}}{2\left(-\frac{1}{2}\right)}$$

-\frac{591}{100} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\frac{3}{10}\right)±\frac{\sqrt{591}i}{10}}{2\left(-\frac{1}{2}\right)}$$

-\frac{3}{10} ning teskarisi \frac{3}{10} ga teng.

$$x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{2\left(-\frac{1}{2}\right)}$$

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

$$x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{-1}$$

x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{-1} tenglamasini yeching, bunda ± musbat. \frac{3}{10} ni \frac{i\sqrt{591}}{10} ga qo'shish.

$$x=\frac{3+\sqrt{591}i}{-10}$$

\frac{3+i\sqrt{591}}{10} ni -1 ga bo'lish.

$$x=\frac{-\sqrt{591}i-3}{10}$$

x=\frac{\frac{3}{10}±\frac{\sqrt{591}i}{10}}{-1} tenglamasini yeching, bunda ± manfiy. \frac{3}{10} dan \frac{i\sqrt{591}}{10} ni ayirish.

$$x=\frac{-\sqrt{591}i+3}{-10}$$

\frac{3-i\sqrt{591}}{10} ni -1 ga bo'lish.

$$x=\frac{-3+\sqrt{591}i}{10}$$

Tenglama yechildi.

$$x=\frac{-\sqrt{591}i-3}{10} x=\frac{-3+\sqrt{591}i}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

5x ga \frac{1}{10}x+\frac{1}{10} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{1}{5}x-3=5x\times \frac{1}{10}x+5x\times \frac{1}{10}$$

x^{2} hosil qilish uchun x va x ni ko'paytirish.

$$\frac{1}{5}x-3=5x^{2}\times \frac{1}{10}+5x\times \frac{1}{10}$$

\frac{5}{10} hosil qilish uchun 5 va \frac{1}{10} ni ko'paytirish.

$$\frac{1}{5}x-3=\frac{5}{10}x^{2}+5x\times \frac{1}{10}$$

\frac{5}{10} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{1}{5}x-3=\frac{1}{2}x^{2}+5x\times \frac{1}{10}$$

\frac{5}{10} hosil qilish uchun 5 va \frac{1}{10} ni ko'paytirish.

$$\frac{1}{5}x-3=\frac{1}{2}x^{2}+\frac{5}{10}x$$

\frac{5}{10} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

Ikkala tarafdan \frac{1}{2}x^{2} ni ayirish.

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

Ikkala tarafdan \frac{1}{2}x ni ayirish.

$$\frac{1}{5}x-3-\frac{1}{2}x^{2}-\frac{1}{2}x=0$$

-\frac{3}{10}x ni olish uchun \frac{1}{5}x va -\frac{1}{2}x ni birlashtirish.

$$-\frac{3}{10}x-3-\frac{1}{2}x^{2}=0$$

3 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$-\frac{3}{10}x-\frac{1}{2}x^{2}=3$$

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

$$-\frac{1}{2}x^{2}-\frac{3}{10}x=3$$

Ikkala tarafini -2 ga ko‘paytiring.

$$\frac{-\frac{1}{2}x^{2}-\frac{3}{10}x}{-\frac{1}{2}}=\frac{3}{-\frac{1}{2}}$$

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

$$x^{2}+\left(-\frac{\frac{3}{10}}{-\frac{1}{2}}\right)x=\frac{3}{-\frac{1}{2}}$$

-\frac{3}{10} ni -\frac{1}{2} ga bo'lish -\frac{3}{10} ga k'paytirish -\frac{1}{2} ga qaytarish.

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

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

$$x^{2}+\frac{3}{5}x=-6$$

\frac{3}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{10} olish uchun. Keyin, \frac{3}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{5}x+\left(\frac{3}{10}\right)^{2}=-6+\left(\frac{3}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{10} kvadratini chiqarish.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-6+\frac{9}{100}$$

-6 ni \frac{9}{100} ga qo'shish.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-\frac{591}{100}$$

x^{2}+\frac{3}{5}x+\frac{9}{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.

$$\left(x+\frac{3}{10}\right)^{2}=-\frac{591}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{10}\right)^{2}}=\sqrt{-\frac{591}{100}}$$

Qisqartirish.

$$x+\frac{3}{10}=\frac{\sqrt{591}i}{10} x+\frac{3}{10}=-\frac{\sqrt{591}i}{10}$$

Tenglamaning ikkala tarafidan \frac{3}{10} ni ayirish.

$$x=\frac{-3+\sqrt{591}i}{10} x=\frac{-\sqrt{591}i-3}{10}$$