1500/x-1500/x+250=1/2 eching

$$\frac { 1500 } { x } - \frac { 1500 } { x + 250 } = \frac { 1 } { 2 }$$

x uchun yechish

$$x=-1000$$

$$x=750$$

Guruhlash orqali faktorlash qadamlari

x qiymati -250,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+250\right) ga, x,x+250,2 ning eng kichik karralisiga ko‘paytiring.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

2x+500 ga 1500 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

3000 hosil qilish uchun 2 va 1500 ni ko'paytirish.

$$3000x+750000-3000x=x\left(x+250\right)$$

x ga x+250 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3000x+750000-3000x=x^{2}+250x$$

Ikkala tarafdan x^{2} ni ayirish.

$$3000x+750000-3000x-x^{2}=250x$$

Ikkala tarafdan 250x ni ayirish.

$$3000x+750000-3000x-x^{2}-250x=0$$

2750x ni olish uchun 3000x va -250x ni birlashtirish.

$$2750x+750000-3000x-x^{2}=0$$

-250x ni olish uchun 2750x va -3000x ni birlashtirish.

$$-250x+750000-x^{2}=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$-x^{2}-250x+750000=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -x^{2}+ax+bx+750000 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-250 ab=-750000=-750000$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -750000-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-750000 2,-375000 3,-250000 4,-187500 5,-150000 6,-125000 8,-93750 10,-75000 12,-62500 15,-50000 16,-46875 20,-37500 24,-31250 25,-30000 30,-25000 40,-18750 48,-15625 50,-15000 60,-12500 75,-10000 80,-9375 100,-7500 120,-6250 125,-6000 150,-5000 200,-3750 240,-3125 250,-3000 300,-2500 375,-2000 400,-1875 500,-1500 600,-1250 625,-1200 750,-1000$$

Har bir juftlik yigʻindisini hisoblang.

$$1-750000=-749999 2-375000=-374998 3-250000=-249997 4-187500=-187496 5-150000=-149995 6-125000=-124994 8-93750=-93742 10-75000=-74990 12-62500=-62488 15-50000=-49985 16-46875=-46859 20-37500=-37480 24-31250=-31226 25-30000=-29975 30-25000=-24970 40-18750=-18710 48-15625=-15577 50-15000=-14950 60-12500=-12440 75-10000=-9925 80-9375=-9295 100-7500=-7400 120-6250=-6130 125-6000=-5875 150-5000=-4850 200-3750=-3550 240-3125=-2885 250-3000=-2750 300-2500=-2200 375-2000=-1625 400-1875=-1475 500-1500=-1000 600-1250=-650 625-1200=-575 750-1000=-250$$

Yechim – 250 yigʻindisini beruvchi juftlik.

$$a=-750 b=1000$$

-x^{2}-250x+750000 ni \left(-x^{2}-750x\right)+\left(1000x+750000\right) sifatida qaytadan yozish.

$$\left(-x^{2}-750x\right)+\left(1000x+750000\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 1000 ni faktordan chiqaring.

$$x\left(x-750\right)+1000\left(x-750\right)$$

Distributiv funktsiyasidan foydalangan holda x-750 umumiy terminini chiqaring.

$$\left(x-750\right)\left(x+1000\right)$$

Tenglamani yechish uchun x-750=0 va x+1000=0 ni yeching.

$$x=750 x=-1000$$

Kvadrat tenglamalaridan foydalanish qoidalari

x qiymati -250,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+250\right) ga, x,x+250,2 ning eng kichik karralisiga ko‘paytiring.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

2x+500 ga 1500 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

3000 hosil qilish uchun 2 va 1500 ni ko'paytirish.

$$3000x+750000-3000x=x\left(x+250\right)$$

x ga x+250 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3000x+750000-3000x=x^{2}+250x$$

Ikkala tarafdan x^{2} ni ayirish.

$$3000x+750000-3000x-x^{2}=250x$$

Ikkala tarafdan 250x ni ayirish.

$$3000x+750000-3000x-x^{2}-250x=0$$

2750x ni olish uchun 3000x va -250x ni birlashtirish.

$$2750x+750000-3000x-x^{2}=0$$

-250x ni olish uchun 2750x va -3000x ni birlashtirish.

$$-250x+750000-x^{2}=0$$

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.

$$-x^{2}-250x+750000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, -250 ni b va 750000 ni c bilan almashtiring.

$$x=\frac{-\left(-250\right)±\sqrt{\left(-250\right)^{2}-4\left(-1\right)\times 750000}}{2\left(-1\right)}$$

-250 kvadratini chiqarish.

$$x=\frac{-\left(-250\right)±\sqrt{62500-4\left(-1\right)\times 750000}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-\left(-250\right)±\sqrt{62500+4\times 750000}}{2\left(-1\right)}$$

4 ni 750000 marotabaga ko'paytirish.

$$x=\frac{-\left(-250\right)±\sqrt{62500+3000000}}{2\left(-1\right)}$$

62500 ni 3000000 ga qo'shish.

$$x=\frac{-\left(-250\right)±\sqrt{3062500}}{2\left(-1\right)}$$

3062500 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-250\right)±1750}{2\left(-1\right)}$$

-250 ning teskarisi 250 ga teng.

$$x=\frac{250±1750}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{250±1750}{-2}$$

x=\frac{250±1750}{-2} tenglamasini yeching, bunda ± musbat. 250 ni 1750 ga qo'shish.

$$x=\frac{2000}{-2}$$

2000 ni -2 ga bo'lish.

$$x=-1000$$

x=\frac{250±1750}{-2} tenglamasini yeching, bunda ± manfiy. 250 dan 1750 ni ayirish.

$$x=-\frac{1500}{-2}$$

-1500 ni -2 ga bo'lish.

$$x=750$$

Tenglama yechildi.

$$x=-1000 x=750$$

Kvadratni yakunlash bo'yicha qadamlar

x qiymati -250,0 qiymatlaridan birortasiga teng bo‘lmaydi, chunki nolga bo‘lish mumkin emas. Tenglamaning ikkala tarafini 2x\left(x+250\right) ga, x,x+250,2 ning eng kichik karralisiga ko‘paytiring.

$$\left(2x+500\right)\times 1500-2x\times 1500=x\left(x+250\right)$$

2x+500 ga 1500 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3000x+750000-2x\times 1500=x\left(x+250\right)$$

3000 hosil qilish uchun 2 va 1500 ni ko'paytirish.

$$3000x+750000-3000x=x\left(x+250\right)$$

x ga x+250 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$3000x+750000-3000x=x^{2}+250x$$

Ikkala tarafdan x^{2} ni ayirish.

$$3000x+750000-3000x-x^{2}=250x$$

Ikkala tarafdan 250x ni ayirish.

$$3000x+750000-3000x-x^{2}-250x=0$$

2750x ni olish uchun 3000x va -250x ni birlashtirish.

$$2750x+750000-3000x-x^{2}=0$$

Ikkala tarafdan 750000 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$2750x-3000x-x^{2}=-750000$$

-250x ni olish uchun 2750x va -3000x ni birlashtirish.

$$-250x-x^{2}=-750000$$

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

$$-x^{2}-250x=-750000$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}-250x}{-1}=-\frac{750000}{-1}$$

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

$$x^{2}+\left(-\frac{250}{-1}\right)x=-\frac{750000}{-1}$$

-250 ni -1 ga bo'lish.

$$x^{2}+250x=-\frac{750000}{-1}$$

-750000 ni -1 ga bo'lish.

$$x^{2}+250x=750000$$

250 ni bo‘lish, x shartining koeffitsienti, 2 ga 125 olish uchun. Keyin, 125 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+250x+125^{2}=750000+125^{2}$$

125 kvadratini chiqarish.

$$x^{2}+250x+15625=750000+15625$$

750000 ni 15625 ga qo'shish.

$$x^{2}+250x+15625=765625$$

x^{2}+250x+15625 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+125\right)^{2}=765625$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+125\right)^{2}}=\sqrt{765625}$$

Qisqartirish.

$$x+125=875 x+125=-875$$

Tenglamaning ikkala tarafidan 125 ni ayirish.

$$x=750 x=-1000$$