x32-x/2=120 eching

$$x \frac{ 32-x }{ 2 } = 120$$

x uchun yechish

$$x=12$$

$$x=20$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$x\left(32-x\right)=240$$

x ga 32-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$32x-x^{2}=240$$

Ikkala tarafdan 240 ni ayirish.

$$32x-x^{2}-240=0$$

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.

$$-x^{2}+32x-240=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -1 ni a, 32 ni b va -240 ni c bilan almashtiring.

$$x=\frac{-32±\sqrt{32^{2}-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

32 kvadratini chiqarish.

$$x=\frac{-32±\sqrt{1024-4\left(-1\right)\left(-240\right)}}{2\left(-1\right)}$$

-4 ni -1 marotabaga ko'paytirish.

$$x=\frac{-32±\sqrt{1024+4\left(-240\right)}}{2\left(-1\right)}$$

4 ni -240 marotabaga ko'paytirish.

$$x=\frac{-32±\sqrt{1024-960}}{2\left(-1\right)}$$

1024 ni -960 ga qo'shish.

$$x=\frac{-32±\sqrt{64}}{2\left(-1\right)}$$

64 ning kvadrat ildizini chiqarish.

$$x=\frac{-32±8}{2\left(-1\right)}$$

2 ni -1 marotabaga ko'paytirish.

$$x=\frac{-32±8}{-2}$$

x=\frac{-32±8}{-2} tenglamasini yeching, bunda ± musbat. -32 ni 8 ga qo'shish.

$$x=-\frac{24}{-2}$$

-24 ni -2 ga bo'lish.

$$x=12$$

x=\frac{-32±8}{-2} tenglamasini yeching, bunda ± manfiy. -32 dan 8 ni ayirish.

$$x=-\frac{40}{-2}$$

-40 ni -2 ga bo'lish.

$$x=20$$

Tenglama yechildi.

$$x=12 x=20$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$x\left(32-x\right)=240$$

x ga 32-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$32x-x^{2}=240$$

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

$$-x^{2}+32x=240$$

Ikki tarafini -1 ga bo‘ling.

$$\frac{-x^{2}+32x}{-1}=\frac{240}{-1}$$

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

$$x^{2}+\frac{32}{-1}x=\frac{240}{-1}$$

32 ni -1 ga bo'lish.

$$x^{2}-32x=\frac{240}{-1}$$

240 ni -1 ga bo'lish.

$$x^{2}-32x=-240$$

-32 ni bo‘lish, x shartining koeffitsienti, 2 ga -16 olish uchun. Keyin, -16 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-32x+\left(-16\right)^{2}=-240+\left(-16\right)^{2}$$

-16 kvadratini chiqarish.

$$x^{2}-32x+256=-240+256$$

-240 ni 256 ga qo'shish.

$$x^{2}-32x+256=16$$

x^{2}-32x+256 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

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

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-16\right)^{2}}=\sqrt{16}$$

Qisqartirish.

$$x-16=4 x-16=-4$$

16 ni tenglamaning ikkala tarafiga qo'shish.

$$x=20 x=12$$