(x(100-x)=500 eching

$$( x ( 100 - x ) = 500$$

x uchun yechish

$$x=20\sqrt{5}+50\approx 94,72135955$$

$$x=50-20\sqrt{5}\approx 5,27864045$$

Kvadrat tenglamalaridan foydalanish qoidalari

x ga 100-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$100x-x^{2}=500$$

Ikkala tarafdan 500 ni ayirish.

$$100x-x^{2}-500=0$$

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.

$$-x^{2}+100x-500=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 100 ni b va -500 ni c bilan almashtiring.

$$x=\frac{-100±\sqrt{100^{2}-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

100 kvadratini chiqarish.

$$x=\frac{-100±\sqrt{10000-4\left(-1\right)\left(-500\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000+4\left(-500\right)}}{2\left(-1\right)}$$

4 ni -500 marotabaga ko'paytirish.

$$x=\frac{-100±\sqrt{10000-2000}}{2\left(-1\right)}$$

10000 ni -2000 ga qo'shish.

$$x=\frac{-100±\sqrt{8000}}{2\left(-1\right)}$$

8000 ning kvadrat ildizini chiqarish.

$$x=\frac{-100±40\sqrt{5}}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-100±40\sqrt{5}}{-2}$$

x=\frac{-100±40\sqrt{5}}{-2} tenglamasini yeching, bunda ± musbat. -100 ni 40\sqrt{5} ga qo'shish.

$$x=\frac{40\sqrt{5}-100}{-2}$$

-100+40\sqrt{5} ni -2 ga bo'lish.

$$x=50-20\sqrt{5}$$

x=\frac{-100±40\sqrt{5}}{-2} tenglamasini yeching, bunda ± manfiy. -100 dan 40\sqrt{5} ni ayirish.

$$x=\frac{-40\sqrt{5}-100}{-2}$$

-100-40\sqrt{5} ni -2 ga bo'lish.

$$x=20\sqrt{5}+50$$

Tenglama yechildi.

$$x=50-20\sqrt{5} x=20\sqrt{5}+50$$

Kvadratni yakunlash bo'yicha qadamlar

x ga 100-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$100x-x^{2}=500$$

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

$$-x^{2}+100x=500$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+100x}{-1}=\frac{500}{-1}$$

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

$$x^{2}+\frac{100}{-1}x=\frac{500}{-1}$$

100 ni -1 ga bo'lish.

$$x^{2}-100x=\frac{500}{-1}$$

500 ni -1 ga bo'lish.

$$x^{2}-100x=-500$$

-100 ni bo‘lish, x shartining koeffitsienti, 2 ga -50 olish uchun. Keyin, -50 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-100x+\left(-50\right)^{2}=-500+\left(-50\right)^{2}$$

-50 kvadratini chiqarish.

$$x^{2}-100x+2500=-500+2500$$

-500 ni 2500 ga qo'shish.

$$x^{2}-100x+2500=2000$$

x^{2}-100x+2500 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-50\right)^{2}=2000$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-50\right)^{2}}=\sqrt{2000}$$

Qisqartirish.

$$x-50=20\sqrt{5} x-50=-20\sqrt{5}$$

50 ni tenglamaning ikkala tarafiga qo'shish.

$$x=20\sqrt{5}+50 x=50-20\sqrt{5}$$