(200-x)(300-5x)=32000 eching

$$(200-x)(300-5x)=32000$$

x uchun yechish

$$x=10\sqrt{113}+130\approx 236,301458127$$

$$x=130-10\sqrt{113}\approx 23,698541873$$

Kvadrat tenglamalaridan foydalanish qoidalari

200-x ga 300-5x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$60000-1300x+5x^{2}=32000$$

Ikkala tarafdan 32000 ni ayirish.

$$60000-1300x+5x^{2}-32000=0$$

28000 olish uchun 60000 dan 32000 ni ayirish.

$$28000-1300x+5x^{2}=0$$

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.

$$5x^{2}-1300x+28000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, -1300 ni b va 28000 ni c bilan almashtiring.

$$x=\frac{-\left(-1300\right)±\sqrt{\left(-1300\right)^{2}-4\times 5\times 28000}}{2\times 5}$$

-1300 kvadratini chiqarish.

$$x=\frac{-\left(-1300\right)±\sqrt{1690000-4\times 5\times 28000}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-\left(-1300\right)±\sqrt{1690000-20\times 28000}}{2\times 5}$$

-20 ni 28000 marotabaga ko'paytirish.

$$x=\frac{-\left(-1300\right)±\sqrt{1690000-560000}}{2\times 5}$$

1690000 ni -560000 ga qo'shish.

$$x=\frac{-\left(-1300\right)±\sqrt{1130000}}{2\times 5}$$

1130000 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-1300\right)±100\sqrt{113}}{2\times 5}$$

-1300 ning teskarisi 1300 ga teng.

$$x=\frac{1300±100\sqrt{113}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{1300±100\sqrt{113}}{10}$$

x=\frac{1300±100\sqrt{113}}{10} tenglamasini yeching, bunda ± musbat. 1300 ni 100\sqrt{113} ga qo'shish.

$$x=\frac{100\sqrt{113}+1300}{10}$$

1300+100\sqrt{113} ni 10 ga bo'lish.

$$x=10\sqrt{113}+130$$

x=\frac{1300±100\sqrt{113}}{10} tenglamasini yeching, bunda ± manfiy. 1300 dan 100\sqrt{113} ni ayirish.

$$x=\frac{1300-100\sqrt{113}}{10}$$

1300-100\sqrt{113} ni 10 ga bo'lish.

$$x=130-10\sqrt{113}$$

Tenglama yechildi.

$$x=10\sqrt{113}+130 x=130-10\sqrt{113}$$

Kvadratni yakunlash bo'yicha qadamlar

200-x ga 300-5x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$60000-1300x+5x^{2}=32000$$

Ikkala tarafdan 60000 ni ayirish.

$$-1300x+5x^{2}=32000-60000$$

-28000 olish uchun 32000 dan 60000 ni ayirish.

$$-1300x+5x^{2}=-28000$$

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

$$5x^{2}-1300x=-28000$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}-1300x}{5}=-\frac{28000}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{1300}{5}\right)x=-\frac{28000}{5}$$

-1300 ni 5 ga bo'lish.

$$x^{2}-260x=-\frac{28000}{5}$$

-28000 ni 5 ga bo'lish.

$$x^{2}-260x=-5600$$

-260 ni bo‘lish, x shartining koeffitsienti, 2 ga -130 olish uchun. Keyin, -130 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-260x+\left(-130\right)^{2}=-5600+\left(-130\right)^{2}$$

-130 kvadratini chiqarish.

$$x^{2}-260x+16900=-5600+16900$$

-5600 ni 16900 ga qo'shish.

$$x^{2}-260x+16900=11300$$

x^{2}-260x+16900 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-130\right)^{2}=11300$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-130\right)^{2}}=\sqrt{11300}$$

Qisqartirish.

$$x-130=10\sqrt{113} x-130=-10\sqrt{113}$$

130 ni tenglamaning ikkala tarafiga qo'shish.

$$x=10\sqrt{113}+130 x=130-10\sqrt{113}$$