800x+500x(9+x)=6000 eching

$$800x+500x(9+x)=6000$$

x uchun yechish

$$x = \frac{\sqrt{4009} - 53}{10} \approx 1,031666447$$

$$x=\frac{-\sqrt{4009}-53}{10}\approx -11,631666447$$

Kvadrat tenglamalaridan foydalanish qoidalari

500x ga 9+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$800x+4500x+500x^{2}=6000$$

5300x ni olish uchun 800x va 4500x ni birlashtirish.

$$5300x+500x^{2}=6000$$

Ikkala tarafdan 6000 ni ayirish.

$$5300x+500x^{2}-6000=0$$

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.

$$500x^{2}+5300x-6000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 500 ni a, 5300 ni b va -6000 ni c bilan almashtiring.

$$x=\frac{-5300±\sqrt{5300^{2}-4\times 500\left(-6000\right)}}{2\times 500}$$

5300 kvadratini chiqarish.

$$x=\frac{-5300±\sqrt{28090000-4\times 500\left(-6000\right)}}{2\times 500}$$

-4 ni 500 marotabaga ko'paytirish.

$$x=\frac{-5300±\sqrt{28090000-2000\left(-6000\right)}}{2\times 500}$$

-2000 ni -6000 marotabaga ko'paytirish.

$$x=\frac{-5300±\sqrt{28090000+12000000}}{2\times 500}$$

28090000 ni 12000000 ga qo'shish.

$$x=\frac{-5300±\sqrt{40090000}}{2\times 500}$$

40090000 ning kvadrat ildizini chiqarish.

$$x=\frac{-5300±100\sqrt{4009}}{2\times 500}$$

2 ni 500 marotabaga ko'paytirish.

$$x=\frac{-5300±100\sqrt{4009}}{1000}$$

x=\frac{-5300±100\sqrt{4009}}{1000} tenglamasini yeching, bunda ± musbat. -5300 ni 100\sqrt{4009} ga qo'shish.

$$x=\frac{100\sqrt{4009}-5300}{1000}$$

-5300+100\sqrt{4009} ni 1000 ga bo'lish.

$$x=\frac{\sqrt{4009}-53}{10}$$

x=\frac{-5300±100\sqrt{4009}}{1000} tenglamasini yeching, bunda ± manfiy. -5300 dan 100\sqrt{4009} ni ayirish.

$$x=\frac{-100\sqrt{4009}-5300}{1000}$$

-5300-100\sqrt{4009} ni 1000 ga bo'lish.

$$x=\frac{-\sqrt{4009}-53}{10}$$

Tenglama yechildi.

$$x=\frac{\sqrt{4009}-53}{10} x=\frac{-\sqrt{4009}-53}{10}$$

Kvadratni yakunlash bo'yicha qadamlar

500x ga 9+x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$800x+4500x+500x^{2}=6000$$

5300x ni olish uchun 800x va 4500x ni birlashtirish.

$$5300x+500x^{2}=6000$$

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

$$500x^{2}+5300x=6000$$

Ikki tarafini 500 ga bo‘ling.

$$\frac{500x^{2}+5300x}{500}=\frac{6000}{500}$$

500 ga bo'lish 500 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{5300}{500}x=\frac{6000}{500}$$

\frac{5300}{500} ulushini 100 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{53}{5}x=\frac{6000}{500}$$

6000 ni 500 ga bo'lish.

$$x^{2}+\frac{53}{5}x=12$$

\frac{53}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{53}{10} olish uchun. Keyin, \frac{53}{10} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{53}{5}x+\left(\frac{53}{10}\right)^{2}=12+\left(\frac{53}{10}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{53}{10} kvadratini chiqarish.

$$x^{2}+\frac{53}{5}x+\frac{2809}{100}=12+\frac{2809}{100}$$

12 ni \frac{2809}{100} ga qo'shish.

$$x^{2}+\frac{53}{5}x+\frac{2809}{100}=\frac{4009}{100}$$

x^{2}+\frac{53}{5}x+\frac{2809}{100} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{53}{10}\right)^{2}=\frac{4009}{100}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{53}{10}\right)^{2}}=\sqrt{\frac{4009}{100}}$$

Qisqartirish.

$$x+\frac{53}{10}=\frac{\sqrt{4009}}{10} x+\frac{53}{10}=-\frac{\sqrt{4009}}{10}$$

Tenglamaning ikkala tarafidan \frac{53}{10} ni ayirish.

$$x=\frac{\sqrt{4009}-53}{10} x=\frac{-\sqrt{4009}-53}{10}$$