(x-3)^2=x/4 eching

$$( x - 3 ) ^ { 2 } = \frac { x } { 4 }$$

x uchun yechish

$$x=4$$

$$x = \frac{9}{4} = 2\frac{1}{4} = 2,25$$

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 4 ga ko'paytirish.

$$4\left(x-3\right)^{2}=x$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-3\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(x^{2}-6x+9\right)=x$$

4 ga x^{2}-6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x$$

Ikkala tarafdan x ni ayirish.

$$4x^{2}-24x+36-x=0$$

-25x ni olish uchun -24x va -x ni birlashtirish.

$$4x^{2}-25x+36=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon 4x^{2}+ax+bx+36 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-25 ab=4\times 36=144$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 144-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

Yechim – -25 yigʻindisini beruvchi juftlik.

$$a=-16 b=-9$$

4x^{2}-25x+36 ni \left(4x^{2}-16x\right)+\left(-9x+36\right) sifatida qaytadan yozish.

$$\left(4x^{2}-16x\right)+\left(-9x+36\right)$$

Birinchi guruhda 4x ni va ikkinchi guruhda -9 ni faktordan chiqaring.

$$4x\left(x-4\right)-9\left(x-4\right)$$

Distributiv funktsiyasidan foydalangan holda x-4 umumiy terminini chiqaring.

$$\left(x-4\right)\left(4x-9\right)$$

Tenglamani yechish uchun x-4=0 va 4x-9=0 ni yeching.

$$x=4 x=\frac{9}{4}$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 4 ga ko'paytirish.

$$4\left(x-3\right)^{2}=x$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-3\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(x^{2}-6x+9\right)=x$$

4 ga x^{2}-6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x$$

Ikkala tarafdan x ni ayirish.

$$4x^{2}-24x+36-x=0$$

-25x ni olish uchun -24x va -x ni birlashtirish.

$$4x^{2}-25x+36=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -25 ni b va 36 ni c bilan almashtiring.

$$x=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 4\times 36}}{2\times 4}$$

-25 kvadratini chiqarish.

$$x=\frac{-\left(-25\right)±\sqrt{625-4\times 4\times 36}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-25\right)±\sqrt{625-16\times 36}}{2\times 4}$$

-16 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-25\right)±\sqrt{625-576}}{2\times 4}$$

625 ni -576 ga qo'shish.

$$x=\frac{-\left(-25\right)±\sqrt{49}}{2\times 4}$$

49 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-25\right)±7}{2\times 4}$$

-25 ning teskarisi 25 ga teng.

$$x=\frac{25±7}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{25±7}{8}$$

x=\frac{25±7}{8} tenglamasini yeching, bunda ± musbat. 25 ni 7 ga qo'shish.

$$x=\frac{32}{8}$$

32 ni 8 ga bo'lish.

$$x=4$$

x=\frac{25±7}{8} tenglamasini yeching, bunda ± manfiy. 25 dan 7 ni ayirish.

$$x=\frac{18}{8}$$

\frac{18}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{9}{4}$$

Tenglama yechildi.

$$x=4 x=\frac{9}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 4 ga ko'paytirish.

$$4\left(x-3\right)^{2}=x$$

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(x-3\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(x^{2}-6x+9\right)=x$$

4 ga x^{2}-6x+9 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4x^{2}-24x+36=x$$

Ikkala tarafdan x ni ayirish.

$$4x^{2}-24x+36-x=0$$

-25x ni olish uchun -24x va -x ni birlashtirish.

$$4x^{2}-25x+36=0$$

Ikkala tarafdan 36 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$4x^{2}-25x=-36$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-25x}{4}=-\frac{36}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{25}{4}x=-\frac{36}{4}$$

-36 ni 4 ga bo'lish.

$$x^{2}-\frac{25}{4}x=-9$$

-\frac{25}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{25}{8} olish uchun. Keyin, -\frac{25}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{25}{4}x+\left(-\frac{25}{8}\right)^{2}=-9+\left(-\frac{25}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{25}{8} kvadratini chiqarish.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=-9+\frac{625}{64}$$

-9 ni \frac{625}{64} ga qo'shish.

$$x^{2}-\frac{25}{4}x+\frac{625}{64}=\frac{49}{64}$$

x^{2}-\frac{25}{4}x+\frac{625}{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.

$$\left(x-\frac{25}{8}\right)^{2}=\frac{49}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{25}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Qisqartirish.

$$x-\frac{25}{8}=\frac{7}{8} x-\frac{25}{8}=-\frac{7}{8}$$

\frac{25}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=4 x=\frac{9}{4}$$