60/x+20+15=100/x eching

$$\frac{ 60 }{ x+20 } +15= \frac{ 100 }{ x }$$

x uchun yechish

$$x = \frac{2 \sqrt{469} - 26}{3} \approx 5,770938552$$

$$x=\frac{-2\sqrt{469}-26}{3}\approx -23,104271885$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -20,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+20\right) ga, x+20,x ning eng kichik karralisiga ko‘paytiring.

$$x\times 60+x\left(x+20\right)\times 15=\left(x+20\right)\times 100$$

x ga x+20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 60+\left(x^{2}+20x\right)\times 15=\left(x+20\right)\times 100$$

x^{2}+20x ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 60+15x^{2}+300x=\left(x+20\right)\times 100$$

360x ni olish uchun x\times 60 va 300x ni birlashtirish.

$$360x+15x^{2}=\left(x+20\right)\times 100$$

x+20 ga 100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+15x^{2}=100x+2000$$

Ikkala tarafdan 100x ni ayirish.

$$360x+15x^{2}-100x=2000$$

260x ni olish uchun 360x va -100x ni birlashtirish.

$$260x+15x^{2}=2000$$

Ikkala tarafdan 2000 ni ayirish.

$$260x+15x^{2}-2000=0$$

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.

$$15x^{2}+260x-2000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, 260 ni b va -2000 ni c bilan almashtiring.

$$x=\frac{-260±\sqrt{260^{2}-4\times 15\left(-2000\right)}}{2\times 15}$$

260 kvadratini chiqarish.

$$x=\frac{-260±\sqrt{67600-4\times 15\left(-2000\right)}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-260±\sqrt{67600-60\left(-2000\right)}}{2\times 15}$$

-60 ni -2000 marotabaga ko'paytirish.

$$x=\frac{-260±\sqrt{67600+120000}}{2\times 15}$$

67600 ni 120000 ga qo'shish.

$$x=\frac{-260±\sqrt{187600}}{2\times 15}$$

187600 ning kvadrat ildizini chiqarish.

$$x=\frac{-260±20\sqrt{469}}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{-260±20\sqrt{469}}{30}$$

x=\frac{-260±20\sqrt{469}}{30} tenglamasini yeching, bunda ± musbat. -260 ni 20\sqrt{469} ga qo'shish.

$$x=\frac{20\sqrt{469}-260}{30}$$

-260+20\sqrt{469} ni 30 ga bo'lish.

$$x=\frac{2\sqrt{469}-26}{3}$$

x=\frac{-260±20\sqrt{469}}{30} tenglamasini yeching, bunda ± manfiy. -260 dan 20\sqrt{469} ni ayirish.

$$x=\frac{-20\sqrt{469}-260}{30}$$

-260-20\sqrt{469} ni 30 ga bo'lish.

$$x=\frac{-2\sqrt{469}-26}{3}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{469}-26}{3} x=\frac{-2\sqrt{469}-26}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -20,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x+20\right) ga, x+20,x ning eng kichik karralisiga ko‘paytiring.

$$x\times 60+x\left(x+20\right)\times 15=\left(x+20\right)\times 100$$

x ga x+20 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 60+\left(x^{2}+20x\right)\times 15=\left(x+20\right)\times 100$$

x^{2}+20x ga 15 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$x\times 60+15x^{2}+300x=\left(x+20\right)\times 100$$

360x ni olish uchun x\times 60 va 300x ni birlashtirish.

$$360x+15x^{2}=\left(x+20\right)\times 100$$

x+20 ga 100 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$360x+15x^{2}=100x+2000$$

Ikkala tarafdan 100x ni ayirish.

$$360x+15x^{2}-100x=2000$$

260x ni olish uchun 360x va -100x ni birlashtirish.

$$260x+15x^{2}=2000$$

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

$$15x^{2}+260x=2000$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15x^{2}+260x}{15}=\frac{2000}{15}$$

15 ga bo'lish 15 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{260}{15}x=\frac{2000}{15}$$

\frac{260}{15} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{52}{3}x=\frac{2000}{15}$$

\frac{2000}{15} ulushini 5 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{52}{3}x=\frac{400}{3}$$

\frac{52}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{26}{3} olish uchun. Keyin, \frac{26}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{52}{3}x+\left(\frac{26}{3}\right)^{2}=\frac{400}{3}+\left(\frac{26}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{26}{3} kvadratini chiqarish.

$$x^{2}+\frac{52}{3}x+\frac{676}{9}=\frac{400}{3}+\frac{676}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{400}{3} ni \frac{676}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{52}{3}x+\frac{676}{9}=\frac{1876}{9}$$

x^{2}+\frac{52}{3}x+\frac{676}{9} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{26}{3}\right)^{2}=\frac{1876}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{26}{3}\right)^{2}}=\sqrt{\frac{1876}{9}}$$

Qisqartirish.

$$x+\frac{26}{3}=\frac{2\sqrt{469}}{3} x+\frac{26}{3}=-\frac{2\sqrt{469}}{3}$$

Tenglamaning ikkala tarafidan \frac{26}{3} ni ayirish.

$$x=\frac{2\sqrt{469}-26}{3} x=\frac{-2\sqrt{469}-26}{3}$$