3600/2500=(1+R/200)^2 eching

$$\frac { 3600 } { 2500 } = ( \frac { 1 + R } { 200 } ) ^ { 2 }$$

R uchun yechish

$$R=239$$

$$R=-241$$

Kvadrat tenglamalaridan foydalanish qoidalari

\frac{3600}{2500} ulushini 100 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{36}{25}=\left(\frac{1+R}{200}\right)^{2}$$

\frac{1+R}{200}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{36}{25}=\frac{\left(1+R\right)^{2}}{200^{2}}$$

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

$$\frac{36}{25}=\frac{1+2R+R^{2}}{200^{2}}$$

2 daraja ko‘rsatkichini 200 ga hisoblang va 40000 ni qiymatni oling.

$$\frac{36}{25}=\frac{1+2R+R^{2}}{40000}$$

\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2} natijani olish uchun 1+2R+R^{2} ning har bir ifodasini 40000 ga bo‘ling.

$$\frac{36}{25}=\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{36}{25}$$

Ikkala tarafdan \frac{36}{25} ni ayirish.

$$\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}-\frac{36}{25}=0$$

-\frac{57599}{40000} olish uchun \frac{1}{40000} dan \frac{36}{25} ni ayirish.

$$-\frac{57599}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}=0$$

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.

$$\frac{1}{40000}R^{2}+\frac{1}{20000}R-\frac{57599}{40000}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{1}{40000} ni a, \frac{1}{20000} ni b va -\frac{57599}{40000} ni c bilan almashtiring.

$$R=\frac{-\frac{1}{20000}±\sqrt{\left(\frac{1}{20000}\right)^{2}-4\times \frac{1}{40000}\left(-\frac{57599}{40000}\right)}}{2\times \frac{1}{40000}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{20000} kvadratini chiqarish.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{1}{400000000}-4\times \frac{1}{40000}\left(-\frac{57599}{40000}\right)}}{2\times \frac{1}{40000}}$$

-4 ni \frac{1}{40000} marotabaga ko'paytirish.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{1}{400000000}-\frac{1}{10000}\left(-\frac{57599}{40000}\right)}}{2\times \frac{1}{40000}}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali -\frac{1}{10000} ni -\frac{57599}{40000} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{1+57599}{400000000}}}{2\times \frac{1}{40000}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{400000000} ni \frac{57599}{400000000} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$R=\frac{-\frac{1}{20000}±\sqrt{\frac{9}{62500}}}{2\times \frac{1}{40000}}$$

\frac{9}{62500} ning kvadrat ildizini chiqarish.

$$R=\frac{-\frac{1}{20000}±\frac{3}{250}}{2\times \frac{1}{40000}}$$

2 ni \frac{1}{40000} marotabaga ko'paytirish.

$$R=\frac{-\frac{1}{20000}±\frac{3}{250}}{\frac{1}{20000}}$$

R=\frac{-\frac{1}{20000}±\frac{3}{250}}{\frac{1}{20000}} tenglamasini yeching, bunda ± musbat. Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{20000} ni \frac{3}{250} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$R=\frac{\frac{239}{20000}}{\frac{1}{20000}}$$

\frac{239}{20000} ni \frac{1}{20000} ga bo'lish \frac{239}{20000} ga k'paytirish \frac{1}{20000} ga qaytarish.

$$R=239$$

R=\frac{-\frac{1}{20000}±\frac{3}{250}}{\frac{1}{20000}} tenglamasini yeching, bunda ± manfiy. Umumiy maxrajni topib va suratlarni ayirib \frac{3}{250} ni -\frac{1}{20000} dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$R=-\frac{\frac{241}{20000}}{\frac{1}{20000}}$$

-\frac{241}{20000} ni \frac{1}{20000} ga bo'lish -\frac{241}{20000} ga k'paytirish \frac{1}{20000} ga qaytarish.

$$R=-241$$

Tenglama yechildi.

$$R=239 R=-241$$

Kvadratni yakunlash bo'yicha qadamlar

\frac{3600}{2500} ulushini 100 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$\frac{36}{25}=\left(\frac{1+R}{200}\right)^{2}$$

\frac{1+R}{200}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$\frac{36}{25}=\frac{\left(1+R\right)^{2}}{200^{2}}$$

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

$$\frac{36}{25}=\frac{1+2R+R^{2}}{200^{2}}$$

2 daraja ko‘rsatkichini 200 ga hisoblang va 40000 ni qiymatni oling.

$$\frac{36}{25}=\frac{1+2R+R^{2}}{40000}$$

\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2} natijani olish uchun 1+2R+R^{2} ning har bir ifodasini 40000 ga bo‘ling.

$$\frac{36}{25}=\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$\frac{1}{40000}+\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{36}{25}$$

Ikkala tarafdan \frac{1}{40000} ni ayirish.

$$\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{36}{25}-\frac{1}{40000}$$

\frac{57599}{40000} olish uchun \frac{36}{25} dan \frac{1}{40000} ni ayirish.

$$\frac{1}{20000}R+\frac{1}{40000}R^{2}=\frac{57599}{40000}$$

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

$$\frac{1}{40000}R^{2}+\frac{1}{20000}R=\frac{57599}{40000}$$

Ikkala tarafini 40000 ga ko‘paytiring.

$$\frac{\frac{1}{40000}R^{2}+\frac{1}{20000}R}{\frac{1}{40000}}=\frac{\frac{57599}{40000}}{\frac{1}{40000}}$$

\frac{1}{40000} ga bo'lish \frac{1}{40000} ga ko'paytirishni bekor qiladi.

$$R^{2}+\frac{\frac{1}{20000}}{\frac{1}{40000}}R=\frac{\frac{57599}{40000}}{\frac{1}{40000}}$$

\frac{1}{20000} ni \frac{1}{40000} ga bo'lish \frac{1}{20000} ga k'paytirish \frac{1}{40000} ga qaytarish.

$$R^{2}+2R=\frac{\frac{57599}{40000}}{\frac{1}{40000}}$$

\frac{57599}{40000} ni \frac{1}{40000} ga bo'lish \frac{57599}{40000} ga k'paytirish \frac{1}{40000} ga qaytarish.

$$R^{2}+2R=57599$$

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

$$R^{2}+2R+1^{2}=57599+1^{2}$$

1 kvadratini chiqarish.

$$R^{2}+2R+1=57599+1$$

57599 ni 1 ga qo'shish.

$$R^{2}+2R+1=57600$$

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

$$\left(R+1\right)^{2}=57600$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(R+1\right)^{2}}=\sqrt{57600}$$

Qisqartirish.

$$R+1=240 R+1=-240$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$R=239 R=-241$$