(6x-11)^2=12x eching

$$( 6 x - 11 ) ^ { 2 } = 12 x$$

x uchun yechish

$$x=\frac{\sqrt{23}}{6}+2\approx 2,799305254$$

$$x=-\frac{\sqrt{23}}{6}+2\approx 1,200694746$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(6x-11\right)^{2} kengaytirilishi uchun ishlating.

$$36x^{2}-132x+121=12x$$

Ikkala tarafdan 12x ni ayirish.

$$36x^{2}-132x+121-12x=0$$

-144x ni olish uchun -132x va -12x ni birlashtirish.

$$36x^{2}-144x+121=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 36 ni a, -144 ni b va 121 ni c bilan almashtiring.

$$x=\frac{-\left(-144\right)±\sqrt{\left(-144\right)^{2}-4\times 36\times 121}}{2\times 36}$$

-144 kvadratini chiqarish.

$$x=\frac{-\left(-144\right)±\sqrt{20736-4\times 36\times 121}}{2\times 36}$$

-4 ni 36 marotabaga ko'paytirish.

$$x=\frac{-\left(-144\right)±\sqrt{20736-144\times 121}}{2\times 36}$$

-144 ni 121 marotabaga ko'paytirish.

$$x=\frac{-\left(-144\right)±\sqrt{20736-17424}}{2\times 36}$$

20736 ni -17424 ga qo'shish.

$$x=\frac{-\left(-144\right)±\sqrt{3312}}{2\times 36}$$

3312 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-144\right)±12\sqrt{23}}{2\times 36}$$

-144 ning teskarisi 144 ga teng.

$$x=\frac{144±12\sqrt{23}}{2\times 36}$$

2 ni 36 marotabaga ko'paytirish.

$$x=\frac{144±12\sqrt{23}}{72}$$

x=\frac{144±12\sqrt{23}}{72} tenglamasini yeching, bunda ± musbat. 144 ni 12\sqrt{23} ga qo'shish.

$$x=\frac{12\sqrt{23}+144}{72}$$

144+12\sqrt{23} ni 72 ga bo'lish.

$$x=\frac{\sqrt{23}}{6}+2$$

x=\frac{144±12\sqrt{23}}{72} tenglamasini yeching, bunda ± manfiy. 144 dan 12\sqrt{23} ni ayirish.

$$x=\frac{144-12\sqrt{23}}{72}$$

144-12\sqrt{23} ni 72 ga bo'lish.

$$x=-\frac{\sqrt{23}}{6}+2$$

Tenglama yechildi.

$$x=\frac{\sqrt{23}}{6}+2 x=-\frac{\sqrt{23}}{6}+2$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(6x-11\right)^{2} kengaytirilishi uchun ishlating.

$$36x^{2}-132x+121=12x$$

Ikkala tarafdan 12x ni ayirish.

$$36x^{2}-132x+121-12x=0$$

-144x ni olish uchun -132x va -12x ni birlashtirish.

$$36x^{2}-144x+121=0$$

Ikkala tarafdan 121 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$36x^{2}-144x=-121$$

Ikki tarafini 36 ga bo‘ling.

$$\frac{36x^{2}-144x}{36}=-\frac{121}{36}$$

36 ga bo'lish 36 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{144}{36}\right)x=-\frac{121}{36}$$

-144 ni 36 ga bo'lish.

$$x^{2}-4x=-\frac{121}{36}$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-\frac{121}{36}+\left(-2\right)^{2}$$

-2 kvadratini chiqarish.

$$x^{2}-4x+4=-\frac{121}{36}+4$$

-\frac{121}{36} ni 4 ga qo'shish.

$$x^{2}-4x+4=\frac{23}{36}$$

x^{2}-4x+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.

$$\left(x-2\right)^{2}=\frac{23}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{\frac{23}{36}}$$

Qisqartirish.

$$x-2=\frac{\sqrt{23}}{6} x-2=-\frac{\sqrt{23}}{6}$$

2 ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{23}}{6}+2 x=-\frac{\sqrt{23}}{6}+2$$