[2/3(x-3)]^2=16*7-x/2 eching

$$[ \frac { 2 } { 3 } ( x - 3 ) ] ^ { 2 } = 16 \times \frac { 7 - x } { 2 }$$

x uchun yechish

$$x=3\sqrt{17}-6\approx 6,369316877$$

$$x=-3\sqrt{17}-6\approx -18,369316877$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$2\times \left(\frac{2}{3}\left(x-3\right)\right)^{2}=16\left(7-x\right)$$

\frac{2}{3} ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2\left(\frac{2}{3}x-2\right)^{2}=16\left(7-x\right)$$

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

$$2\left(\frac{4}{9}x^{2}-\frac{8}{3}x+4\right)=16\left(7-x\right)$$

2 ga \frac{4}{9}x^{2}-\frac{8}{3}x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=16\left(7-x\right)$$

16 ga 7-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=112-16x$$

Ikkala tarafdan 112 ni ayirish.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8-112=-16x$$

-104 olish uchun 8 dan 112 ni ayirish.

$$\frac{8}{9}x^{2}-\frac{16}{3}x-104=-16x$$

16x ni ikki tarafga qo’shing.

$$\frac{8}{9}x^{2}-\frac{16}{3}x-104+16x=0$$

\frac{32}{3}x ni olish uchun -\frac{16}{3}x va 16x ni birlashtirish.

$$\frac{8}{9}x^{2}+\frac{32}{3}x-104=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \frac{8}{9} ni a, \frac{32}{3} ni b va -104 ni c bilan almashtiring.

$$x=\frac{-\frac{32}{3}±\sqrt{\left(\frac{32}{3}\right)^{2}-4\times \frac{8}{9}\left(-104\right)}}{2\times \frac{8}{9}}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{32}{3} kvadratini chiqarish.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{1024}{9}-4\times \frac{8}{9}\left(-104\right)}}{2\times \frac{8}{9}}$$

-4 ni \frac{8}{9} marotabaga ko'paytirish.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{1024}{9}-\frac{32}{9}\left(-104\right)}}{2\times \frac{8}{9}}$$

-\frac{32}{9} ni -104 marotabaga ko'paytirish.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{1024+3328}{9}}}{2\times \frac{8}{9}}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1024}{9} ni \frac{3328}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x=\frac{-\frac{32}{3}±\sqrt{\frac{4352}{9}}}{2\times \frac{8}{9}}$$

\frac{4352}{9} ning kvadrat ildizini chiqarish.

$$x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{2\times \frac{8}{9}}$$

2 ni \frac{8}{9} marotabaga ko'paytirish.

$$x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{\frac{16}{9}}$$

x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{\frac{16}{9}} tenglamasini yeching, bunda ± musbat. -\frac{32}{3} ni \frac{16\sqrt{17}}{3} ga qo'shish.

$$x=\frac{16\sqrt{17}-32}{\frac{16}{9}\times 3}$$

\frac{-32+16\sqrt{17}}{3} ni \frac{16}{9} ga bo'lish \frac{-32+16\sqrt{17}}{3} ga k'paytirish \frac{16}{9} ga qaytarish.

$$x=3\sqrt{17}-6$$

x=\frac{-\frac{32}{3}±\frac{16\sqrt{17}}{3}}{\frac{16}{9}} tenglamasini yeching, bunda ± manfiy. -\frac{32}{3} dan \frac{16\sqrt{17}}{3} ni ayirish.

$$x=\frac{-16\sqrt{17}-32}{\frac{16}{9}\times 3}$$

\frac{-32-16\sqrt{17}}{3} ni \frac{16}{9} ga bo'lish \frac{-32-16\sqrt{17}}{3} ga k'paytirish \frac{16}{9} ga qaytarish.

$$x=-3\sqrt{17}-6$$

Tenglama yechildi.

$$x=3\sqrt{17}-6 x=-3\sqrt{17}-6$$

Kvadratni yakunlash bo'yicha qadamlar

Tenglamaning ikkala tarafini 2 ga ko'paytirish.

$$2\times \left(\frac{2}{3}\left(x-3\right)\right)^{2}=16\left(7-x\right)$$

\frac{2}{3} ga x-3 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$2\left(\frac{2}{3}x-2\right)^{2}=16\left(7-x\right)$$

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

$$2\left(\frac{4}{9}x^{2}-\frac{8}{3}x+4\right)=16\left(7-x\right)$$

2 ga \frac{4}{9}x^{2}-\frac{8}{3}x+4 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=16\left(7-x\right)$$

16 ga 7-x ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8=112-16x$$

16x ni ikki tarafga qo’shing.

$$\frac{8}{9}x^{2}-\frac{16}{3}x+8+16x=112$$

\frac{32}{3}x ni olish uchun -\frac{16}{3}x va 16x ni birlashtirish.

$$\frac{8}{9}x^{2}+\frac{32}{3}x+8=112$$

Ikkala tarafdan 8 ni ayirish.

$$\frac{8}{9}x^{2}+\frac{32}{3}x=112-8$$

104 olish uchun 112 dan 8 ni ayirish.

$$\frac{8}{9}x^{2}+\frac{32}{3}x=104$$

Tenglamaning ikki tarafini \frac{8}{9} ga bo'lish, bu kasrni qaytarish orqali ikkala tarafga ko'paytirish bilan aynidir.

$$\frac{\frac{8}{9}x^{2}+\frac{32}{3}x}{\frac{8}{9}}=\frac{104}{\frac{8}{9}}$$

\frac{8}{9} ga bo'lish \frac{8}{9} ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{\frac{32}{3}}{\frac{8}{9}}x=\frac{104}{\frac{8}{9}}$$

\frac{32}{3} ni \frac{8}{9} ga bo'lish \frac{32}{3} ga k'paytirish \frac{8}{9} ga qaytarish.

$$x^{2}+12x=\frac{104}{\frac{8}{9}}$$

104 ni \frac{8}{9} ga bo'lish 104 ga k'paytirish \frac{8}{9} ga qaytarish.

$$x^{2}+12x=117$$

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.

$$x^{2}+12x+6^{2}=117+6^{2}$$

6 kvadratini chiqarish.

$$x^{2}+12x+36=117+36$$

117 ni 36 ga qo'shish.

$$x^{2}+12x+36=153$$

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.

$$\left(x+6\right)^{2}=153$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{153}$$

Qisqartirish.

$$x+6=3\sqrt{17} x+6=-3\sqrt{17}$$

Tenglamaning ikkala tarafidan 6 ni ayirish.

$$x=3\sqrt{17}-6 x=-3\sqrt{17}-6$$