1050/x+1050/x-10=42 eching

$$\frac { 1050 } { x } + \frac { 1050 } { x - 10 } = 42$$

x uchun yechish

$$x=5\sqrt{26}+30\approx 55,495097568$$

$$x=30-5\sqrt{26}\approx 4,504902432$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati 0,10 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-10\right) ga, x,x-10 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-10\right)\times 1050+x\times 1050=42x\left(x-10\right)$$

x-10 ga 1050 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$1050x-10500+x\times 1050=42x\left(x-10\right)$$

2100x ni olish uchun 1050x va x\times 1050 ni birlashtirish.

$$2100x-10500=42x\left(x-10\right)$$

42x ga x-10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2100x-10500=42x^{2}-420x$$

Ikkala tarafdan 42x^{2} ni ayirish.

$$2100x-10500-42x^{2}=-420x$$

420x ni ikki tarafga qo’shing.

$$2100x-10500-42x^{2}+420x=0$$

2520x ni olish uchun 2100x va 420x ni birlashtirish.

$$2520x-10500-42x^{2}=0$$

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.

$$-42x^{2}+2520x-10500=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -42 ni a, 2520 ni b va -10500 ni c bilan almashtiring.

$$x=\frac{-2520±\sqrt{2520^{2}-4\left(-42\right)\left(-10500\right)}}{2\left(-42\right)}$$

2520 kvadratini chiqarish.

$$x=\frac{-2520±\sqrt{6350400-4\left(-42\right)\left(-10500\right)}}{2\left(-42\right)}$$

-4 ni -42 marotabaga ko'paytirish.

$$x=\frac{-2520±\sqrt{6350400+168\left(-10500\right)}}{2\left(-42\right)}$$

168 ni -10500 marotabaga ko'paytirish.

$$x=\frac{-2520±\sqrt{6350400-1764000}}{2\left(-42\right)}$$

6350400 ni -1764000 ga qo'shish.

$$x=\frac{-2520±\sqrt{4586400}}{2\left(-42\right)}$$

4586400 ning kvadrat ildizini chiqarish.

$$x=\frac{-2520±420\sqrt{26}}{2\left(-42\right)}$$

2 ni -42 marotabaga ko'paytirish.

$$x=\frac{-2520±420\sqrt{26}}{-84}$$

x=\frac{-2520±420\sqrt{26}}{-84} tenglamasini yeching, bunda ± musbat. -2520 ni 420\sqrt{26} ga qo'shish.

$$x=\frac{420\sqrt{26}-2520}{-84}$$

-2520+420\sqrt{26} ni -84 ga bo'lish.

$$x=30-5\sqrt{26}$$

x=\frac{-2520±420\sqrt{26}}{-84} tenglamasini yeching, bunda ± manfiy. -2520 dan 420\sqrt{26} ni ayirish.

$$x=\frac{-420\sqrt{26}-2520}{-84}$$

-2520-420\sqrt{26} ni -84 ga bo'lish.

$$x=5\sqrt{26}+30$$

Tenglama yechildi.

$$x=30-5\sqrt{26} x=5\sqrt{26}+30$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati 0,10 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini x\left(x-10\right) ga, x,x-10 ning eng kichik karralisiga ko‘paytiring.

$$\left(x-10\right)\times 1050+x\times 1050=42x\left(x-10\right)$$

x-10 ga 1050 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$1050x-10500+x\times 1050=42x\left(x-10\right)$$

2100x ni olish uchun 1050x va x\times 1050 ni birlashtirish.

$$2100x-10500=42x\left(x-10\right)$$

42x ga x-10 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2100x-10500=42x^{2}-420x$$

Ikkala tarafdan 42x^{2} ni ayirish.

$$2100x-10500-42x^{2}=-420x$$

420x ni ikki tarafga qo’shing.

$$2100x-10500-42x^{2}+420x=0$$

2520x ni olish uchun 2100x va 420x ni birlashtirish.

$$2520x-10500-42x^{2}=0$$

10500 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$2520x-42x^{2}=10500$$

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

$$-42x^{2}+2520x=10500$$

Ikki tarafini -42 ga bo‘ling.

$$\frac{-42x^{2}+2520x}{-42}=\frac{10500}{-42}$$

-42 ga bo'lish -42 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2520}{-42}x=\frac{10500}{-42}$$

2520 ni -42 ga bo'lish.

$$x^{2}-60x=\frac{10500}{-42}$$

10500 ni -42 ga bo'lish.

$$x^{2}-60x=-250$$

-60 ni bo‘lish, x shartining koeffitsienti, 2 ga -30 olish uchun. Keyin, -30 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-60x+\left(-30\right)^{2}=-250+\left(-30\right)^{2}$$

-30 kvadratini chiqarish.

$$x^{2}-60x+900=-250+900$$

-250 ni 900 ga qo'shish.

$$x^{2}-60x+900=650$$

x^{2}-60x+900 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-30\right)^{2}=650$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-30\right)^{2}}=\sqrt{650}$$

Qisqartirish.

$$x-30=5\sqrt{26} x-30=-5\sqrt{26}$$

30 ni tenglamaning ikkala tarafiga qo'shish.

$$x=5\sqrt{26}+30 x=30-5\sqrt{26}$$