70/x+35+70/x-35=40 eching

$$\frac{ 70 }{ x+35 } + \frac{ 70 }{ x-35 } =40$$

x uchun yechish

$$x = \frac{7 \sqrt{401} + 7}{4} \approx 36,79372269$$

$$x=\frac{7-7\sqrt{401}}{4}\approx -33,29372269$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -35,35 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-35\right)\left(x+35\right) ga, x+35,x-35 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-35\right)\times 70+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

x-35 ga 70 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$70x-2450+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

x+35 ga 70 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$70x-2450+70x+2450=40\left(x-35\right)\left(x+35\right)$$

140x ni olish uchun 70x va 70x ni birlashtirish.

$$140x-2450+2450=40\left(x-35\right)\left(x+35\right)$$

0 olish uchun -2450 va 2450'ni qo'shing.

$$140x=40\left(x-35\right)\left(x+35\right)$$

40 ga x-35 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$140x=\left(40x-1400\right)\left(x+35\right)$$

40x-1400 ga x+35 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$140x=40x^{2}-49000$$

Ikkala tarafdan 40x^{2} ni ayirish.

$$140x-40x^{2}=-49000$$

49000 ni ikki tarafga qo’shing.

$$140x-40x^{2}+49000=0$$

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.

$$-40x^{2}+140x+49000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -40 ni a, 140 ni b va 49000 ni c bilan almashtiring.

$$x=\frac{-140±\sqrt{140^{2}-4\left(-40\right)\times 49000}}{2\left(-40\right)}$$

140 kvadratini chiqarish.

$$x=\frac{-140±\sqrt{19600-4\left(-40\right)\times 49000}}{2\left(-40\right)}$$

-4 ni -40 marotabaga ko'paytirish.

$$x=\frac{-140±\sqrt{19600+160\times 49000}}{2\left(-40\right)}$$

160 ni 49000 marotabaga ko'paytirish.

$$x=\frac{-140±\sqrt{19600+7840000}}{2\left(-40\right)}$$

19600 ni 7840000 ga qo'shish.

$$x=\frac{-140±\sqrt{7859600}}{2\left(-40\right)}$$

7859600 ning kvadrat ildizini chiqarish.

$$x=\frac{-140±140\sqrt{401}}{2\left(-40\right)}$$

2 ni -40 marotabaga ko'paytirish.

$$x=\frac{-140±140\sqrt{401}}{-80}$$

x=\frac{-140±140\sqrt{401}}{-80} tenglamasini yeching, bunda ± musbat. -140 ni 140\sqrt{401} ga qo'shish.

$$x=\frac{140\sqrt{401}-140}{-80}$$

-140+140\sqrt{401} ni -80 ga bo'lish.

$$x=\frac{7-7\sqrt{401}}{4}$$

x=\frac{-140±140\sqrt{401}}{-80} tenglamasini yeching, bunda ± manfiy. -140 dan 140\sqrt{401} ni ayirish.

$$x=\frac{-140\sqrt{401}-140}{-80}$$

-140-140\sqrt{401} ni -80 ga bo'lish.

$$x=\frac{7\sqrt{401}+7}{4}$$

Tenglama yechildi.

$$x=\frac{7-7\sqrt{401}}{4} x=\frac{7\sqrt{401}+7}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -35,35 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini \left(x-35\right)\left(x+35\right) ga, x+35,x-35 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-35\right)\times 70+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

x-35 ga 70 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$70x-2450+\left(x+35\right)\times 70=40\left(x-35\right)\left(x+35\right)$$

x+35 ga 70 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$70x-2450+70x+2450=40\left(x-35\right)\left(x+35\right)$$

140x ni olish uchun 70x va 70x ni birlashtirish.

$$140x-2450+2450=40\left(x-35\right)\left(x+35\right)$$

0 olish uchun -2450 va 2450'ni qo'shing.

$$140x=40\left(x-35\right)\left(x+35\right)$$

40 ga x-35 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$140x=\left(40x-1400\right)\left(x+35\right)$$

40x-1400 ga x+35 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$140x=40x^{2}-49000$$

Ikkala tarafdan 40x^{2} ni ayirish.

$$140x-40x^{2}=-49000$$

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

$$-40x^{2}+140x=-49000$$

Ikki tarafini -40 ga bo‘ling.

$$\frac{-40x^{2}+140x}{-40}=-\frac{49000}{-40}$$

-40 ga bo'lish -40 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{140}{-40}x=-\frac{49000}{-40}$$

\frac{140}{-40} ulushini 20 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{7}{2}x=-\frac{49000}{-40}$$

-49000 ni -40 ga bo'lish.

$$x^{2}-\frac{7}{2}x=1225$$

-\frac{7}{2} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{7}{4} olish uchun. Keyin, -\frac{7}{4} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{7}{2}x+\left(-\frac{7}{4}\right)^{2}=1225+\left(-\frac{7}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{7}{4} kvadratini chiqarish.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=1225+\frac{49}{16}$$

1225 ni \frac{49}{16} ga qo'shish.

$$x^{2}-\frac{7}{2}x+\frac{49}{16}=\frac{19649}{16}$$

x^{2}-\frac{7}{2}x+\frac{49}{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.

$$\left(x-\frac{7}{4}\right)^{2}=\frac{19649}{16}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{7}{4}\right)^{2}}=\sqrt{\frac{19649}{16}}$$

Qisqartirish.

$$x-\frac{7}{4}=\frac{7\sqrt{401}}{4} x-\frac{7}{4}=-\frac{7\sqrt{401}}{4}$$

\frac{7}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{7\sqrt{401}+7}{4} x=\frac{7-7\sqrt{401}}{4}$$