left(12-xright)^2+144=9x^2 eching

$${ \left(12-x \right) }^{ 2 } +144=9 { x }^{ 2 }$$

x uchun yechish

$$x = \frac{3 \sqrt{17} - 3}{2} \approx 4,684658438$$

$$x=\frac{-3\sqrt{17}-3}{2}\approx -7,684658438$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(12-x\right)^{2} kengaytirilishi uchun ishlating.

$$144-24x+x^{2}+144=9x^{2}$$

288 olish uchun 144 va 144'ni qo'shing.

$$288-24x+x^{2}=9x^{2}$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$288-24x+x^{2}-9x^{2}=0$$

-8x^{2} ni olish uchun x^{2} va -9x^{2} ni birlashtirish.

$$288-24x-8x^{2}=0$$

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.

$$-8x^{2}-24x+288=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -8 ni a, -24 ni b va 288 ni c bilan almashtiring.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-8\right)\times 288}}{2\left(-8\right)}$$

-24 kvadratini chiqarish.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\left(-8\right)\times 288}}{2\left(-8\right)}$$

-4 ni -8 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576+32\times 288}}{2\left(-8\right)}$$

32 ni 288 marotabaga ko'paytirish.

$$x=\frac{-\left(-24\right)±\sqrt{576+9216}}{2\left(-8\right)}$$

576 ni 9216 ga qo'shish.

$$x=\frac{-\left(-24\right)±\sqrt{9792}}{2\left(-8\right)}$$

9792 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-24\right)±24\sqrt{17}}{2\left(-8\right)}$$

-24 ning teskarisi 24 ga teng.

$$x=\frac{24±24\sqrt{17}}{2\left(-8\right)}$$

2 ni -8 marotabaga ko'paytirish.

$$x=\frac{24±24\sqrt{17}}{-16}$$

x=\frac{24±24\sqrt{17}}{-16} tenglamasini yeching, bunda ± musbat. 24 ni 24\sqrt{17} ga qo'shish.

$$x=\frac{24\sqrt{17}+24}{-16}$$

24+24\sqrt{17} ni -16 ga bo'lish.

$$x=\frac{-3\sqrt{17}-3}{2}$$

x=\frac{24±24\sqrt{17}}{-16} tenglamasini yeching, bunda ± manfiy. 24 dan 24\sqrt{17} ni ayirish.

$$x=\frac{24-24\sqrt{17}}{-16}$$

24-24\sqrt{17} ni -16 ga bo'lish.

$$x=\frac{3\sqrt{17}-3}{2}$$

Tenglama yechildi.

$$x=\frac{-3\sqrt{17}-3}{2} x=\frac{3\sqrt{17}-3}{2}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(12-x\right)^{2} kengaytirilishi uchun ishlating.

$$144-24x+x^{2}+144=9x^{2}$$

288 olish uchun 144 va 144'ni qo'shing.

$$288-24x+x^{2}=9x^{2}$$

Ikkala tarafdan 9x^{2} ni ayirish.

$$288-24x+x^{2}-9x^{2}=0$$

-8x^{2} ni olish uchun x^{2} va -9x^{2} ni birlashtirish.

$$288-24x-8x^{2}=0$$

Ikkala tarafdan 288 ni ayirish. Har qanday sonni noldan ayirsangiz, o‘zining manfiyi chiqadi.

$$-24x-8x^{2}=-288$$

Bu kabi kvadrat tenglamalarni kvadratni yakunlab yechish mumkin. Kvadratni yechish uchun tenglama avval ushbu shaklda bo'lishi shart: x^{2}+bx=c.

$$-8x^{2}-24x=-288$$

Ikki tarafini -8 ga bo‘ling.

$$\frac{-8x^{2}-24x}{-8}=-\frac{288}{-8}$$

-8 ga bo'lish -8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{24}{-8}\right)x=-\frac{288}{-8}$$

-24 ni -8 ga bo'lish.

$$x^{2}+3x=-\frac{288}{-8}$$

-288 ni -8 ga bo'lish.

$$x^{2}+3x=36$$

3 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{2} olish uchun. Keyin, \frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=36+\left(\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{2} kvadratini chiqarish.

$$x^{2}+3x+\frac{9}{4}=36+\frac{9}{4}$$

36 ni \frac{9}{4} ga qo'shish.

$$x^{2}+3x+\frac{9}{4}=\frac{153}{4}$$

x^{2}+3x+\frac{9}{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.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{153}{4}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{153}{4}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{3\sqrt{17}}{2} x+\frac{3}{2}=-\frac{3\sqrt{17}}{2}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=\frac{3\sqrt{17}-3}{2} x=\frac{-3\sqrt{17}-3}{2}$$