3x^2+72=33x eching

$$3 { x }^{ 2 } +72=33x$$

x uchun yechish

$$x=3$$

$$x=8$$

Guruhlash orqali faktorlash qadamlari

Ikkala tarafdan 33x ni ayirish.

$$3x^{2}+72-33x=0$$

Ikki tarafini 3 ga bo‘ling.

$$x^{2}+24-11x=0$$

Polinomni standart shaklga keltirish uchun uni qayta tartiblang. Shartlarni eng yuqoridan eng pastki qiymat ko'rsatgichiga joylashtirish.

$$x^{2}-11x+24=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx+24 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-11 ab=1\times 24=24$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 24-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

Yechim – -11 yigʻindisini beruvchi juftlik.

$$a=-8 b=-3$$

x^{2}-11x+24 ni \left(x^{2}-8x\right)+\left(-3x+24\right) sifatida qaytadan yozish.

$$\left(x^{2}-8x\right)+\left(-3x+24\right)$$

Birinchi guruhda x ni va ikkinchi guruhda -3 ni faktordan chiqaring.

$$x\left(x-8\right)-3\left(x-8\right)$$

Distributiv funktsiyasidan foydalangan holda x-8 umumiy terminini chiqaring.

$$\left(x-8\right)\left(x-3\right)$$

Tenglamani yechish uchun x-8=0 va x-3=0 ni yeching.

$$x=8 x=3$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 33x ni ayirish.

$$3x^{2}+72-33x=0$$

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.

$$3x^{2}-33x+72=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -33 ni b va 72 ni c bilan almashtiring.

$$x=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 3\times 72}}{2\times 3}$$

-33 kvadratini chiqarish.

$$x=\frac{-\left(-33\right)±\sqrt{1089-4\times 3\times 72}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-33\right)±\sqrt{1089-12\times 72}}{2\times 3}$$

-12 ni 72 marotabaga ko'paytirish.

$$x=\frac{-\left(-33\right)±\sqrt{1089-864}}{2\times 3}$$

1089 ni -864 ga qo'shish.

$$x=\frac{-\left(-33\right)±\sqrt{225}}{2\times 3}$$

225 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-33\right)±15}{2\times 3}$$

-33 ning teskarisi 33 ga teng.

$$x=\frac{33±15}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{33±15}{6}$$

x=\frac{33±15}{6} tenglamasini yeching, bunda ± musbat. 33 ni 15 ga qo'shish.

$$x=\frac{48}{6}$$

48 ni 6 ga bo'lish.

$$x=8$$

x=\frac{33±15}{6} tenglamasini yeching, bunda ± manfiy. 33 dan 15 ni ayirish.

$$x=\frac{18}{6}$$

18 ni 6 ga bo'lish.

$$x=3$$

Tenglama yechildi.

$$x=8 x=3$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 33x ni ayirish.

$$3x^{2}+72-33x=0$$

Ikkala tarafdan 72 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$3x^{2}-33x=-72$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-33x}{3}=-\frac{72}{3}$$

3 ga bo'lish 3 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{33}{3}\right)x=-\frac{72}{3}$$

-33 ni 3 ga bo'lish.

$$x^{2}-11x=-\frac{72}{3}$$

-72 ni 3 ga bo'lish.

$$x^{2}-11x=-24$$

-11 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{2} olish uchun. Keyin, -\frac{11}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=-24+\left(-\frac{11}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{2} kvadratini chiqarish.

$$x^{2}-11x+\frac{121}{4}=-24+\frac{121}{4}$$

-24 ni \frac{121}{4} ga qo'shish.

$$x^{2}-11x+\frac{121}{4}=\frac{25}{4}$$

x^{2}-11x+\frac{121}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{25}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Qisqartirish.

$$x-\frac{11}{2}=\frac{5}{2} x-\frac{11}{2}=-\frac{5}{2}$$

\frac{11}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=8 x=3$$