x(16-x)-120=0 eching

$$x ( 16 - x ) - 120 = 0$$

x uchun yechish (complex solution)

$$x=-2\sqrt{14}i+8\approx 8-7,483314774i$$

$$x=8+2\sqrt{14}i\approx 8+7,483314774i$$

Kvadrat tenglamalaridan foydalanish qoidalari

x ga 16-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-x^{2}-120=0$$

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.

$$-x^{2}+16x-120=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 16 ni b va -120 ni c bilan almashtiring.

$$x=\frac{-16±\sqrt{16^{2}-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

16 kvadratini chiqarish.

$$x=\frac{-16±\sqrt{256-4\left(-1\right)\left(-120\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256+4\left(-120\right)}}{2\left(-1\right)}$$

4 ni -120 marotabaga ko'paytirish.

$$x=\frac{-16±\sqrt{256-480}}{2\left(-1\right)}$$

256 ni -480 ga qo'shish.

$$x=\frac{-16±\sqrt{-224}}{2\left(-1\right)}$$

-224 ning kvadrat ildizini chiqarish.

$$x=\frac{-16±4\sqrt{14}i}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-16±4\sqrt{14}i}{-2}$$

x=\frac{-16±4\sqrt{14}i}{-2} tenglamasini yeching, bunda ± musbat. -16 ni 4i\sqrt{14} ga qo'shish.

$$x=\frac{-16+4\sqrt{14}i}{-2}$$

-16+4i\sqrt{14} ni -2 ga bo'lish.

$$x=-2\sqrt{14}i+8$$

x=\frac{-16±4\sqrt{14}i}{-2} tenglamasini yeching, bunda ± manfiy. -16 dan 4i\sqrt{14} ni ayirish.

$$x=\frac{-4\sqrt{14}i-16}{-2}$$

-16-4i\sqrt{14} ni -2 ga bo'lish.

$$x=8+2\sqrt{14}i$$

Tenglama yechildi.

$$x=-2\sqrt{14}i+8 x=8+2\sqrt{14}i$$

Kvadratni yakunlash bo'yicha qadamlar

x ga 16-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$16x-x^{2}-120=0$$

120 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$16x-x^{2}=120$$

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

$$-x^{2}+16x=120$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+16x}{-1}=\frac{120}{-1}$$

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

$$x^{2}+\frac{16}{-1}x=\frac{120}{-1}$$

16 ni -1 ga bo'lish.

$$x^{2}-16x=\frac{120}{-1}$$

120 ni -1 ga bo'lish.

$$x^{2}-16x=-120$$

-16 ni bo‘lish, x shartining koeffitsienti, 2 ga -8 olish uchun. Keyin, -8 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-16x+\left(-8\right)^{2}=-120+\left(-8\right)^{2}$$

-8 kvadratini chiqarish.

$$x^{2}-16x+64=-120+64$$

-120 ni 64 ga qo'shish.

$$x^{2}-16x+64=-56$$

x^{2}-16x+64 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-8\right)^{2}=-56$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-8\right)^{2}}=\sqrt{-56}$$

Qisqartirish.

$$x-8=2\sqrt{14}i x-8=-2\sqrt{14}i$$

8 ni tenglamaning ikkala tarafiga qo'shish.

$$x=8+2\sqrt{14}i x=-2\sqrt{14}i+8$$