(x-100)[300+(200-x))=3200 eching

$$( x - 100 ) [ 300 + ( 200 - x ) ) = 3200$$

x uchun yechish

$$x=40\sqrt{23}+300\approx 491,833260933$$

$$x=300-40\sqrt{23}\approx 108,166739067$$

Kvadrat tenglamalaridan foydalanish qoidalari

500 olish uchun 300 va 200'ni qo'shing.

$$\left(x-100\right)\left(500-x\right)=3200$$

x-100 ga 500-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$600x-x^{2}-50000=3200$$

Ikkala tarafdan 3200 ni ayirish.

$$600x-x^{2}-50000-3200=0$$

-53200 olish uchun -50000 dan 3200 ni ayirish.

$$600x-x^{2}-53200=0$$

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.

$$-x^{2}+600x-53200=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 600 ni b va -53200 ni c bilan almashtiring.

$$x=\frac{-600±\sqrt{600^{2}-4\left(-1\right)\left(-53200\right)}}{2\left(-1\right)}$$

600 kvadratini chiqarish.

$$x=\frac{-600±\sqrt{360000-4\left(-1\right)\left(-53200\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-600±\sqrt{360000+4\left(-53200\right)}}{2\left(-1\right)}$$

4 ni -53200 marotabaga ko'paytirish.

$$x=\frac{-600±\sqrt{360000-212800}}{2\left(-1\right)}$$

360000 ni -212800 ga qo'shish.

$$x=\frac{-600±\sqrt{147200}}{2\left(-1\right)}$$

147200 ning kvadrat ildizini chiqarish.

$$x=\frac{-600±80\sqrt{23}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-600±80\sqrt{23}}{-2}$$

x=\frac{-600±80\sqrt{23}}{-2} tenglamasini yeching, bunda ± musbat. -600 ni 80\sqrt{23} ga qo'shish.

$$x=\frac{80\sqrt{23}-600}{-2}$$

-600+80\sqrt{23} ni -2 ga bo'lish.

$$x=300-40\sqrt{23}$$

x=\frac{-600±80\sqrt{23}}{-2} tenglamasini yeching, bunda ± manfiy. -600 dan 80\sqrt{23} ni ayirish.

$$x=\frac{-80\sqrt{23}-600}{-2}$$

-600-80\sqrt{23} ni -2 ga bo'lish.

$$x=40\sqrt{23}+300$$

Tenglama yechildi.

$$x=300-40\sqrt{23} x=40\sqrt{23}+300$$

Kvadratni yakunlash bo'yicha qadamlar

500 olish uchun 300 va 200'ni qo'shing.

$$\left(x-100\right)\left(500-x\right)=3200$$

x-100 ga 500-x ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$600x-x^{2}-50000=3200$$

50000 ni ikki tarafga qo’shing.

$$600x-x^{2}=3200+50000$$

53200 olish uchun 3200 va 50000'ni qo'shing.

$$600x-x^{2}=53200$$

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

$$-x^{2}+600x=53200$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+600x}{-1}=\frac{53200}{-1}$$

-1 ga bo'lish -1 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{600}{-1}x=\frac{53200}{-1}$$

600 ni -1 ga bo'lish.

$$x^{2}-600x=\frac{53200}{-1}$$

53200 ni -1 ga bo'lish.

$$x^{2}-600x=-53200$$

-600 ni bo‘lish, x shartining koeffitsienti, 2 ga -300 olish uchun. Keyin, -300 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-600x+\left(-300\right)^{2}=-53200+\left(-300\right)^{2}$$

-300 kvadratini chiqarish.

$$x^{2}-600x+90000=-53200+90000$$

-53200 ni 90000 ga qo'shish.

$$x^{2}-600x+90000=36800$$

x^{2}-600x+90000 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-300\right)^{2}=36800$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-300\right)^{2}}=\sqrt{36800}$$

Qisqartirish.

$$x-300=40\sqrt{23} x-300=-40\sqrt{23}$$

300 ni tenglamaning ikkala tarafiga qo'shish.

$$x=40\sqrt{23}+300 x=300-40\sqrt{23}$$