0.042=(1-x)(1-x)/(1+x) eching

$$0.042 = \frac { ( 1 - x ) ( 1 - x ) } { ( 1 + x ) }$$

x uchun yechish

$$x = \frac{\sqrt{84441} + 1021}{1000} \approx 1,311587336$$

$$x=\frac{1021-\sqrt{84441}}{1000}\approx 0,730412664$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+1 ga ko'paytirish.

$$0,042\left(x+1\right)=\left(1-x\right)\left(1-x\right)$$

\left(1-x\right)^{2} hosil qilish uchun 1-x va 1-x ni ko'paytirish.

$$0,042\left(x+1\right)=\left(1-x\right)^{2}$$

0,042 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$0,042x+0,042=\left(1-x\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(1-x\right)^{2} kengaytirilishi uchun ishlating.

$$0,042x+0,042=1-2x+x^{2}$$

Ikkala tarafdan 1 ni ayirish.

$$0,042x+0,042-1=-2x+x^{2}$$

-0,958 olish uchun 0,042 dan 1 ni ayirish.

$$0,042x-0,958=-2x+x^{2}$$

2x ni ikki tarafga qo’shing.

$$0,042x-0,958+2x=x^{2}$$

2,042x ni olish uchun 0,042x va 2x ni birlashtirish.

$$2,042x-0,958=x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$2,042x-0,958-x^{2}=0$$

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.

$$-x^{2}+2,042x-0,958=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 2,042 ni b va -0,958 ni c bilan almashtiring.

$$x=\frac{-2,042±\sqrt{2,042^{2}-4\left(-1\right)\left(-0,958\right)}}{2\left(-1\right)}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib 2,042 kvadratini chiqarish.

$$x=\frac{-2,042±\sqrt{4,169764-4\left(-1\right)\left(-0,958\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2,042±\sqrt{4,169764+4\left(-0,958\right)}}{2\left(-1\right)}$$

4 ni -0,958 marotabaga ko'paytirish.

$$x=\frac{-2,042±\sqrt{4,169764-3,832}}{2\left(-1\right)}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali 4,169764 ni -3,832 ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-2,042±\sqrt{0,337764}}{2\left(-1\right)}$$

0,337764 ning kvadrat ildizini chiqarish.

$$x=\frac{-2,042±\frac{\sqrt{84441}}{500}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-2,042±\frac{\sqrt{84441}}{500}}{-2}$$

x=\frac{-2,042±\frac{\sqrt{84441}}{500}}{-2} tenglamasini yeching, bunda ± musbat. -2,042 ni \frac{\sqrt{84441}}{500} ga qo'shish.

$$x=\frac{\sqrt{84441}-1021}{-2\times 500}$$

\frac{-1021+\sqrt{84441}}{500} ni -2 ga bo'lish.

$$x=\frac{1021-\sqrt{84441}}{1000}$$

x=\frac{-2,042±\frac{\sqrt{84441}}{500}}{-2} tenglamasini yeching, bunda ± manfiy. -2,042 dan \frac{\sqrt{84441}}{500} ni ayirish.

$$x=\frac{-\sqrt{84441}-1021}{-2\times 500}$$

\frac{-1021-\sqrt{84441}}{500} ni -2 ga bo'lish.

$$x=\frac{\sqrt{84441}+1021}{1000}$$

Tenglama yechildi.

$$x=\frac{1021-\sqrt{84441}}{1000} x=\frac{\sqrt{84441}+1021}{1000}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -1 teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x+1 ga ko'paytirish.

$$0.042\left(x+1\right)=\left(1-x\right)\left(1-x\right)$$

\left(1-x\right)^{2} hosil qilish uchun 1-x va 1-x ni ko'paytirish.

$$0.042\left(x+1\right)=\left(1-x\right)^{2}$$

0.042 ga x+1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$0.042x+0.042=\left(1-x\right)^{2}$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(1-x\right)^{2} kengaytirilishi uchun ishlating.

$$0.042x+0.042=1-2x+x^{2}$$

2x ni ikki tarafga qo’shing.

$$0.042x+0.042+2x=1+x^{2}$$

2.042x ni olish uchun 0.042x va 2x ni birlashtirish.

$$2.042x+0.042=1+x^{2}$$

Ikkala tarafdan x^{2} ni ayirish.

$$2.042x+0.042-x^{2}=1$$

Ikkala tarafdan 0.042 ni ayirish.

$$2.042x-x^{2}=1-0.042$$

0.958 olish uchun 1 dan 0.042 ni ayirish.

$$2.042x-x^{2}=0.958$$

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

$$-x^{2}+2.042x=0.958$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+2.042x}{-1}=\frac{0.958}{-1}$$

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

$$x^{2}+\frac{2.042}{-1}x=\frac{0.958}{-1}$$

2.042 ni -1 ga bo'lish.

$$x^{2}-2.042x=\frac{0.958}{-1}$$

0.958 ni -1 ga bo'lish.

$$x^{2}-2.042x=-0.958$$

-2.042 ni bo‘lish, x shartining koeffitsienti, 2 ga -1.021 olish uchun. Keyin, -1.021 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-2.042x+\left(-1.021\right)^{2}=-0.958+\left(-1.021\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -1.021 kvadratini chiqarish.

$$x^{2}-2.042x+1.042441=-0.958+1.042441$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -0.958 ni 1.042441 ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-2.042x+1.042441=0.084441$$

x^{2}-2.042x+1.042441 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-1.021\right)^{2}=0.084441$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-1.021\right)^{2}}=\sqrt{0.084441}$$

Qisqartirish.

$$x-1.021=\frac{\sqrt{84441}}{1000} x-1.021=-\frac{\sqrt{84441}}{1000}$$

1.021 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{84441}+1021}{1000} x=\frac{1021-\sqrt{84441}}{1000}$$