(100)^2+(x+100)^2=(2x+100)^2 eching

$$( 100 ) ^ { 2 } + ( x + 100 ) ^ { 2 } = ( 2 x + 100 ) ^ { 2 }$$

x uchun yechish

$$x = \frac{100}{3} = 33\frac{1}{3} \approx 33,333333333$$

$$x=-100$$

Guruhlash orqali faktorlash qadamlari

2 daraja ko‘rsatkichini 100 ga hisoblang va 10000 ni qiymatni oling.

$$10000+\left(x+100\right)^{2}=\left(2x+100\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+100\right)^{2} kengaytirilishi uchun ishlating.

$$10000+x^{2}+200x+10000=\left(2x+100\right)^{2}$$

20000 olish uchun 10000 va 10000'ni qo'shing.

$$20000+x^{2}+200x=\left(2x+100\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+100\right)^{2} kengaytirilishi uchun ishlating.

$$20000+x^{2}+200x=4x^{2}+400x+10000$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$20000+x^{2}+200x-4x^{2}=400x+10000$$

-3x^{2} ni olish uchun x^{2} va -4x^{2} ni birlashtirish.

$$20000-3x^{2}+200x=400x+10000$$

Ikkala tarafdan 400x ni ayirish.

$$20000-3x^{2}+200x-400x=10000$$

-200x ni olish uchun 200x va -400x ni birlashtirish.

$$20000-3x^{2}-200x=10000$$

Ikkala tarafdan 10000 ni ayirish.

$$20000-3x^{2}-200x-10000=0$$

10000 olish uchun 20000 dan 10000 ni ayirish.

$$10000-3x^{2}-200x=0$$

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

$$-3x^{2}-200x+10000=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -3x^{2}+ax+bx+10000 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-200 ab=-3\times 10000=-30000$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b manfiy boʻlganda, manfiy sonda musbatga nisbatdan kattaroq mutlaq qiymat bor. -30000-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,-30000 2,-15000 3,-10000 4,-7500 5,-6000 6,-5000 8,-3750 10,-3000 12,-2500 15,-2000 16,-1875 20,-1500 24,-1250 25,-1200 30,-1000 40,-750 48,-625 50,-600 60,-500 75,-400 80,-375 100,-300 120,-250 125,-240 150,-200$$

Har bir juftlik yigʻindisini hisoblang.

$$1-30000=-29999 2-15000=-14998 3-10000=-9997 4-7500=-7496 5-6000=-5995 6-5000=-4994 8-3750=-3742 10-3000=-2990 12-2500=-2488 15-2000=-1985 16-1875=-1859 20-1500=-1480 24-1250=-1226 25-1200=-1175 30-1000=-970 40-750=-710 48-625=-577 50-600=-550 60-500=-440 75-400=-325 80-375=-295 100-300=-200 120-250=-130 125-240=-115 150-200=-50$$

Yechim – -200 yigʻindisini beruvchi juftlik.

$$a=100 b=-300$$

-3x^{2}-200x+10000 ni \left(-3x^{2}+100x\right)+\left(-300x+10000\right) sifatida qaytadan yozish.

$$\left(-3x^{2}+100x\right)+\left(-300x+10000\right)$$

Birinchi guruhda -x ni va ikkinchi guruhda -100 ni faktordan chiqaring.

$$-x\left(3x-100\right)-100\left(3x-100\right)$$

Distributiv funktsiyasidan foydalangan holda 3x-100 umumiy terminini chiqaring.

$$\left(3x-100\right)\left(-x-100\right)$$

Tenglamani yechish uchun 3x-100=0 va -x-100=0 ni yeching.

$$x=\frac{100}{3} x=-100$$

Kvadrat tenglamalaridan foydalanish qoidalari

2 daraja ko‘rsatkichini 100 ga hisoblang va 10000 ni qiymatni oling.

$$10000+\left(x+100\right)^{2}=\left(2x+100\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+100\right)^{2} kengaytirilishi uchun ishlating.

$$10000+x^{2}+200x+10000=\left(2x+100\right)^{2}$$

20000 olish uchun 10000 va 10000'ni qo'shing.

$$20000+x^{2}+200x=\left(2x+100\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+100\right)^{2} kengaytirilishi uchun ishlating.

$$20000+x^{2}+200x=4x^{2}+400x+10000$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$20000+x^{2}+200x-4x^{2}=400x+10000$$

-3x^{2} ni olish uchun x^{2} va -4x^{2} ni birlashtirish.

$$20000-3x^{2}+200x=400x+10000$$

Ikkala tarafdan 400x ni ayirish.

$$20000-3x^{2}+200x-400x=10000$$

-200x ni olish uchun 200x va -400x ni birlashtirish.

$$20000-3x^{2}-200x=10000$$

Ikkala tarafdan 10000 ni ayirish.

$$20000-3x^{2}-200x-10000=0$$

10000 olish uchun 20000 dan 10000 ni ayirish.

$$10000-3x^{2}-200x=0$$

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.

$$-3x^{2}-200x+10000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -200 ni b va 10000 ni c bilan almashtiring.

$$x=\frac{-\left(-200\right)±\sqrt{\left(-200\right)^{2}-4\left(-3\right)\times 10000}}{2\left(-3\right)}$$

-200 kvadratini chiqarish.

$$x=\frac{-\left(-200\right)±\sqrt{40000-4\left(-3\right)\times 10000}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-200\right)±\sqrt{40000+12\times 10000}}{2\left(-3\right)}$$

12 ni 10000 marotabaga ko'paytirish.

$$x=\frac{-\left(-200\right)±\sqrt{40000+120000}}{2\left(-3\right)}$$

40000 ni 120000 ga qo'shish.

$$x=\frac{-\left(-200\right)±\sqrt{160000}}{2\left(-3\right)}$$

160000 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-200\right)±400}{2\left(-3\right)}$$

-200 ning teskarisi 200 ga teng.

$$x=\frac{200±400}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{200±400}{-6}$$

x=\frac{200±400}{-6} tenglamasini yeching, bunda ± musbat. 200 ni 400 ga qo'shish.

$$x=\frac{600}{-6}$$

600 ni -6 ga bo'lish.

$$x=-100$$

x=\frac{200±400}{-6} tenglamasini yeching, bunda ± manfiy. 200 dan 400 ni ayirish.

$$x=-\frac{200}{-6}$$

\frac{-200}{-6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{100}{3}$$

Tenglama yechildi.

$$x=-100 x=\frac{100}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

2 daraja ko‘rsatkichini 100 ga hisoblang va 10000 ni qiymatni oling.

$$10000+\left(x+100\right)^{2}=\left(2x+100\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(x+100\right)^{2} kengaytirilishi uchun ishlating.

$$10000+x^{2}+200x+10000=\left(2x+100\right)^{2}$$

20000 olish uchun 10000 va 10000'ni qo'shing.

$$20000+x^{2}+200x=\left(2x+100\right)^{2}$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(2x+100\right)^{2} kengaytirilishi uchun ishlating.

$$20000+x^{2}+200x=4x^{2}+400x+10000$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$20000+x^{2}+200x-4x^{2}=400x+10000$$

-3x^{2} ni olish uchun x^{2} va -4x^{2} ni birlashtirish.

$$20000-3x^{2}+200x=400x+10000$$

Ikkala tarafdan 400x ni ayirish.

$$20000-3x^{2}+200x-400x=10000$$

-200x ni olish uchun 200x va -400x ni birlashtirish.

$$20000-3x^{2}-200x=10000$$

Ikkala tarafdan 20000 ni ayirish.

$$-3x^{2}-200x=10000-20000$$

-10000 olish uchun 10000 dan 20000 ni ayirish.

$$-3x^{2}-200x=-10000$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}-200x}{-3}=-\frac{10000}{-3}$$

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{200}{-3}\right)x=-\frac{10000}{-3}$$

-200 ni -3 ga bo'lish.

$$x^{2}+\frac{200}{3}x=-\frac{10000}{-3}$$

-10000 ni -3 ga bo'lish.

$$x^{2}+\frac{200}{3}x=\frac{10000}{3}$$

\frac{200}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{100}{3} olish uchun. Keyin, \frac{100}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{200}{3}x+\left(\frac{100}{3}\right)^{2}=\frac{10000}{3}+\left(\frac{100}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{100}{3} kvadratini chiqarish.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{10000}{3}+\frac{10000}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{10000}{3} ni \frac{10000}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{200}{3}x+\frac{10000}{9}=\frac{40000}{9}$$

x^{2}+\frac{200}{3}x+\frac{10000}{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.

$$\left(x+\frac{100}{3}\right)^{2}=\frac{40000}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{100}{3}\right)^{2}}=\sqrt{\frac{40000}{9}}$$

Qisqartirish.

$$x+\frac{100}{3}=\frac{200}{3} x+\frac{100}{3}=-\frac{200}{3}$$

Tenglamaning ikkala tarafidan \frac{100}{3} ni ayirish.

$$x=\frac{100}{3} x=-100$$