(1000-5x)(x-100)=32000 eching

$$(1000-5x)(x-100)=32000$$

x uchun yechish (complex solution)

$$x=150+10\sqrt{39}i\approx 150+62,449979984i$$

$$x=-10\sqrt{39}i+150\approx 150-62,449979984i$$

Kvadrat tenglamalaridan foydalanish qoidalari

1000-5x ga x-100 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$1500x-100000-5x^{2}=32000$$

Ikkala tarafdan 32000 ni ayirish.

$$1500x-100000-5x^{2}-32000=0$$

-132000 olish uchun -100000 dan 32000 ni ayirish.

$$1500x-132000-5x^{2}=0$$

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.

$$-5x^{2}+1500x-132000=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -5 ni a, 1500 ni b va -132000 ni c bilan almashtiring.

$$x=\frac{-1500±\sqrt{1500^{2}-4\left(-5\right)\left(-132000\right)}}{2\left(-5\right)}$$

1500 kvadratini chiqarish.

$$x=\frac{-1500±\sqrt{2250000-4\left(-5\right)\left(-132000\right)}}{2\left(-5\right)}$$

-4 ni -5 marotabaga ko'paytirish.

$$x=\frac{-1500±\sqrt{2250000+20\left(-132000\right)}}{2\left(-5\right)}$$

20 ni -132000 marotabaga ko'paytirish.

$$x=\frac{-1500±\sqrt{2250000-2640000}}{2\left(-5\right)}$$

2250000 ni -2640000 ga qo'shish.

$$x=\frac{-1500±\sqrt{-390000}}{2\left(-5\right)}$$

-390000 ning kvadrat ildizini chiqarish.

$$x=\frac{-1500±100\sqrt{39}i}{2\left(-5\right)}$$

2 ni -5 marotabaga ko'paytirish.

$$x=\frac{-1500±100\sqrt{39}i}{-10}$$

x=\frac{-1500±100\sqrt{39}i}{-10} tenglamasini yeching, bunda ± musbat. -1500 ni 100i\sqrt{39} ga qo'shish.

$$x=\frac{-1500+100\sqrt{39}i}{-10}$$

-1500+100i\sqrt{39} ni -10 ga bo'lish.

$$x=-10\sqrt{39}i+150$$

x=\frac{-1500±100\sqrt{39}i}{-10} tenglamasini yeching, bunda ± manfiy. -1500 dan 100i\sqrt{39} ni ayirish.

$$x=\frac{-100\sqrt{39}i-1500}{-10}$$

-1500-100i\sqrt{39} ni -10 ga bo'lish.

$$x=150+10\sqrt{39}i$$

Tenglama yechildi.

$$x=-10\sqrt{39}i+150 x=150+10\sqrt{39}i$$

Kvadratni yakunlash bo'yicha qadamlar

1000-5x ga x-100 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$1500x-100000-5x^{2}=32000$$

100000 ni ikki tarafga qo’shing.

$$1500x-5x^{2}=32000+100000$$

132000 olish uchun 32000 va 100000'ni qo'shing.

$$1500x-5x^{2}=132000$$

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

$$-5x^{2}+1500x=132000$$

Ikki tarafini -5 ga bo‘ling.

$$\frac{-5x^{2}+1500x}{-5}=\frac{132000}{-5}$$

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

$$x^{2}+\frac{1500}{-5}x=\frac{132000}{-5}$$

1500 ni -5 ga bo'lish.

$$x^{2}-300x=\frac{132000}{-5}$$

132000 ni -5 ga bo'lish.

$$x^{2}-300x=-26400$$

-300 ni bo‘lish, x shartining koeffitsienti, 2 ga -150 olish uchun. Keyin, -150 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-300x+\left(-150\right)^{2}=-26400+\left(-150\right)^{2}$$

-150 kvadratini chiqarish.

$$x^{2}-300x+22500=-26400+22500$$

-26400 ni 22500 ga qo'shish.

$$x^{2}-300x+22500=-3900$$

x^{2}-300x+22500 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-150\right)^{2}=-3900$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-150\right)^{2}}=\sqrt{-3900}$$

Qisqartirish.

$$x-150=10\sqrt{39}i x-150=-10\sqrt{39}i$$

150 ni tenglamaning ikkala tarafiga qo'shish.

$$x=150+10\sqrt{39}i x=-10\sqrt{39}i+150$$