(4sqrt{3}+frac{xsqrt{3}}{2})^2+x^2/4=156 eching

$$( 4 \sqrt { 3 } + \frac { x \sqrt { 3 } } { 2 } ) ^ { 2 } + \frac { x ^ { 2 } } { 4 } = 156$$

x uchun yechish

$$x=-18$$

$$x=6$$

Guruhlash orqali faktorlash qadamlari

Tenglamaning ikkala tarafini 4 ga ko'paytirish.

$$4\left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}+x^{2}=624$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(16\left(\sqrt{3}\right)^{2}+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$4\left(16\times 3+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

48 hosil qilish uchun 16 va 3 ni ko'paytirish.

$$4\left(48+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

8 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

\frac{x\sqrt{3}}{2}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 48 ni \frac{2^{2}}{2^{2}} marotabaga ko'paytirish.

$$4\left(\frac{48\times 2^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

\frac{48\times 2^{2}}{2^{2}} va \frac{\left(x\sqrt{3}\right)^{2}}{2^{2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$4\left(\frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}\right)+x^{2}=624$$

4 ga \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4\times \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4\times \frac{48\times 4+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

192 hosil qilish uchun 48 va 4 ni ko'paytirish.

$$4\times \frac{192+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\left(x\sqrt{3}\right)^{2} ni kengaytirish.

$$4\times \frac{192+x^{2}\left(\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$4\times \frac{192+x^{2}\times 3}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4\times \frac{192+x^{2}\times 3}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

4\times \frac{192+x^{2}\times 3}{4} ni yagona kasrga aylantiring.

$$\frac{4\left(192+x^{2}\times 3\right)}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

4 va 4 ni qisqartiring.

$$192+x^{2}\times 3+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$192+x^{2}\times 3+16\times 3x+x^{2}=624$$

48 hosil qilish uchun 16 va 3 ni ko'paytirish.

$$192+x^{2}\times 3+48x+x^{2}=624$$

4x^{2} ni olish uchun x^{2}\times 3 va x^{2} ni birlashtirish.

$$192+4x^{2}+48x=624$$

Ikkala tarafdan 624 ni ayirish.

$$192+4x^{2}+48x-624=0$$

-432 olish uchun 192 dan 624 ni ayirish.

$$-432+4x^{2}+48x=0$$

Ikki tarafini 4 ga bo‘ling.

$$-108+x^{2}+12x=0$$

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

$$x^{2}+12x-108=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon x^{2}+ax+bx-108 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=12 ab=1\left(-108\right)=-108$$

ab manfiy boʻlganda, a va b da qarama-qarshi belgilar bor. a+b musbat boʻlganda, musbat sonda manfiyga nisbatdan kattaroq mutlaq qiymat bor. -108-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,108 -2,54 -3,36 -4,27 -6,18 -9,12$$

Har bir juftlik yigʻindisini hisoblang.

$$-1+108=107 -2+54=52 -3+36=33 -4+27=23 -6+18=12 -9+12=3$$

Yechim – 12 yigʻindisini beruvchi juftlik.

$$a=-6 b=18$$

x^{2}+12x-108 ni \left(x^{2}-6x\right)+\left(18x-108\right) sifatida qaytadan yozish.

$$\left(x^{2}-6x\right)+\left(18x-108\right)$$

Birinchi guruhda x ni va ikkinchi guruhda 18 ni faktordan chiqaring.

$$x\left(x-6\right)+18\left(x-6\right)$$

Distributiv funktsiyasidan foydalangan holda x-6 umumiy terminini chiqaring.

$$\left(x-6\right)\left(x+18\right)$$

Tenglamani yechish uchun x-6=0 va x+18=0 ni yeching.

$$x=6 x=-18$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 4 ga ko'paytirish.

$$4\left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}+x^{2}=624$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(16\left(\sqrt{3}\right)^{2}+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$4\left(16\times 3+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

48 hosil qilish uchun 16 va 3 ni ko'paytirish.

$$4\left(48+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

8 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

\frac{x\sqrt{3}}{2}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 48 ni \frac{2^{2}}{2^{2}} marotabaga ko'paytirish.

$$4\left(\frac{48\times 2^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

\frac{48\times 2^{2}}{2^{2}} va \frac{\left(x\sqrt{3}\right)^{2}}{2^{2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$4\left(\frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}\right)+x^{2}=624$$

4 ga \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4\times \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4\times \frac{48\times 4+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

192 hosil qilish uchun 48 va 4 ni ko'paytirish.

$$4\times \frac{192+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\left(x\sqrt{3}\right)^{2} ni kengaytirish.

$$4\times \frac{192+x^{2}\left(\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$4\times \frac{192+x^{2}\times 3}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4\times \frac{192+x^{2}\times 3}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

4\times \frac{192+x^{2}\times 3}{4} ni yagona kasrga aylantiring.

$$\frac{4\left(192+x^{2}\times 3\right)}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

4 va 4 ni qisqartiring.

$$192+x^{2}\times 3+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$192+x^{2}\times 3+16\times 3x+x^{2}=624$$

48 hosil qilish uchun 16 va 3 ni ko'paytirish.

$$192+x^{2}\times 3+48x+x^{2}=624$$

4x^{2} ni olish uchun x^{2}\times 3 va x^{2} ni birlashtirish.

$$192+4x^{2}+48x=624$$

Ikkala tarafdan 624 ni ayirish.

$$192+4x^{2}+48x-624=0$$

-432 olish uchun 192 dan 624 ni ayirish.

$$-432+4x^{2}+48x=0$$

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.

$$4x^{2}+48x-432=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, 48 ni b va -432 ni c bilan almashtiring.

$$x=\frac{-48±\sqrt{48^{2}-4\times 4\left(-432\right)}}{2\times 4}$$

48 kvadratini chiqarish.

$$x=\frac{-48±\sqrt{2304-4\times 4\left(-432\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-48±\sqrt{2304-16\left(-432\right)}}{2\times 4}$$

-16 ni -432 marotabaga ko'paytirish.

$$x=\frac{-48±\sqrt{2304+6912}}{2\times 4}$$

2304 ni 6912 ga qo'shish.

$$x=\frac{-48±\sqrt{9216}}{2\times 4}$$

9216 ning kvadrat ildizini chiqarish.

$$x=\frac{-48±96}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{-48±96}{8}$$

x=\frac{-48±96}{8} tenglamasini yeching, bunda ± musbat. -48 ni 96 ga qo'shish.

$$x=\frac{48}{8}$$

48 ni 8 ga bo'lish.

$$x=6$$

x=\frac{-48±96}{8} tenglamasini yeching, bunda ± manfiy. -48 dan 96 ni ayirish.

$$x=-\frac{144}{8}$$

-144 ni 8 ga bo'lish.

$$x=-18$$

Tenglama yechildi.

$$x=6 x=-18$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 4 ga ko'paytirish.

$$4\left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2}+x^{2}=624$$

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(4\sqrt{3}+\frac{x\sqrt{3}}{2}\right)^{2} kengaytirilishi uchun ishlating.

$$4\left(16\left(\sqrt{3}\right)^{2}+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$4\left(16\times 3+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

48 hosil qilish uchun 16 va 3 ni ko'paytirish.

$$4\left(48+8\sqrt{3}\times \frac{x\sqrt{3}}{2}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

8 va 2 ichida eng katta umumiy 2 faktorini bekor qiling.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\left(\frac{x\sqrt{3}}{2}\right)^{2}\right)+x^{2}=624$$

\frac{x\sqrt{3}}{2}ni darajaga oshirish uchun, surat va maxrajni darajaga oshirib, keyin bo‘ling.

$$4\left(48+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. 48 ni \frac{2^{2}}{2^{2}} marotabaga ko'paytirish.

$$4\left(\frac{48\times 2^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}+\frac{\left(x\sqrt{3}\right)^{2}}{2^{2}}\right)+x^{2}=624$$

\frac{48\times 2^{2}}{2^{2}} va \frac{\left(x\sqrt{3}\right)^{2}}{2^{2}} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$4\left(\frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3}\right)+x^{2}=624$$

4 ga \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+4x\sqrt{3}\sqrt{3} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$4\times \frac{48\times 2^{2}+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4\times \frac{48\times 4+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

192 hosil qilish uchun 48 va 4 ni ko'paytirish.

$$4\times \frac{192+\left(x\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\left(x\sqrt{3}\right)^{2} ni kengaytirish.

$$4\times \frac{192+x^{2}\left(\sqrt{3}\right)^{2}}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$4\times \frac{192+x^{2}\times 3}{2^{2}}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

2 daraja ko‘rsatkichini 2 ga hisoblang va 4 ni qiymatni oling.

$$4\times \frac{192+x^{2}\times 3}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

4\times \frac{192+x^{2}\times 3}{4} ni yagona kasrga aylantiring.

$$\frac{4\left(192+x^{2}\times 3\right)}{4}+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

4 va 4 ni qisqartiring.

$$192+x^{2}\times 3+16\left(\sqrt{3}\right)^{2}x+x^{2}=624$$

\sqrt{3} kvadrati – 3.

$$192+x^{2}\times 3+16\times 3x+x^{2}=624$$

48 hosil qilish uchun 16 va 3 ni ko'paytirish.

$$192+x^{2}\times 3+48x+x^{2}=624$$

4x^{2} ni olish uchun x^{2}\times 3 va x^{2} ni birlashtirish.

$$192+4x^{2}+48x=624$$

Ikkala tarafdan 192 ni ayirish.

$$4x^{2}+48x=624-192$$

432 olish uchun 624 dan 192 ni ayirish.

$$4x^{2}+48x=432$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}+48x}{4}=\frac{432}{4}$$

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

$$x^{2}+\frac{48}{4}x=\frac{432}{4}$$

48 ni 4 ga bo'lish.

$$x^{2}+12x=\frac{432}{4}$$

432 ni 4 ga bo'lish.

$$x^{2}+12x=108$$

12 ni bo‘lish, x shartining koeffitsienti, 2 ga 6 olish uchun. Keyin, 6 ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+12x+6^{2}=108+6^{2}$$

6 kvadratini chiqarish.

$$x^{2}+12x+36=108+36$$

108 ni 36 ga qo'shish.

$$x^{2}+12x+36=144$$

x^{2}+12x+36 omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+6\right)^{2}=144$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{144}$$

Qisqartirish.

$$x+6=12 x+6=-12$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$x=6 x=-18$$