(30+x)(1000-3x)-30000-310x=30000 eching

$$(30+x)(1000-3x)-30000-310x=30000$$

x uchun yechish

$$x=100$$

Kvadrat tenglamalaridan foydalanish qoidalari

30+x ga 1000-3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30000+910x-3x^{2}-30000-310x=30000$$

0 olish uchun 30000 dan 30000 ni ayirish.

$$910x-3x^{2}-310x=30000$$

600x ni olish uchun 910x va -310x ni birlashtirish.

$$600x-3x^{2}=30000$$

Ikkala tarafdan 30000 ni ayirish.

$$600x-3x^{2}-30000=0$$

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.

$$-3x^{2}+600x-30000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, 600 ni b va -30000 ni c bilan almashtiring.

$$x=\frac{-600±\sqrt{600^{2}-4\left(-3\right)\left(-30000\right)}}{2\left(-3\right)}$$

600 kvadratini chiqarish.

$$x=\frac{-600±\sqrt{360000-4\left(-3\right)\left(-30000\right)}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-600±\sqrt{360000+12\left(-30000\right)}}{2\left(-3\right)}$$

12 ni -30000 marotabaga ko'paytirish.

$$x=\frac{-600±\sqrt{360000-360000}}{2\left(-3\right)}$$

360000 ni -360000 ga qo'shish.

$$x=\frac{-600±\sqrt{0}}{2\left(-3\right)}$$

0 ning kvadrat ildizini chiqarish.

$$x=-\frac{600}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

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

-600 ni -6 ga bo'lish.

$$x=100$$

Kvadratni yakunlash bo'yicha qadamlar

30+x ga 1000-3x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$30000+910x-3x^{2}-30000-310x=30000$$

0 olish uchun 30000 dan 30000 ni ayirish.

$$910x-3x^{2}-310x=30000$$

600x ni olish uchun 910x va -310x ni birlashtirish.

$$600x-3x^{2}=30000$$

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

$$-3x^{2}+600x=30000$$

Ikki tarafini -3 ga bo‘ling.

$$\frac{-3x^{2}+600x}{-3}=\frac{30000}{-3}$$

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

$$x^{2}+\frac{600}{-3}x=\frac{30000}{-3}$$

600 ni -3 ga bo'lish.

$$x^{2}-200x=\frac{30000}{-3}$$

30000 ni -3 ga bo'lish.

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

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

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

-100 kvadratini chiqarish.

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

-10000 ni 10000 ga qo'shish.

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

x^{2}-200x+10000 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-100\right)^{2}=0$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-100\right)^{2}}=\sqrt{0}$$

Qisqartirish.

$$x-100=0 x-100=0$$

100 ni tenglamaning ikkala tarafiga qo'shish.

$$x=100 x=100$$

Tenglama yechildi. Yechimlar bir xil.

$$x=100$$