sqrt{4578}x^2-sqrt{4677521}x+31478=10523 eching

$$\sqrt{ 4578 } { x }^{ 2 } - \sqrt{ 4677521 } x+31478=10523$$

x uchun yechish (complex solution)

$$x=\frac{\sqrt{21413691138}+i\sqrt{383727960\sqrt{4578}-21413691138}}{9156}\approx 15,982325934+7,366910212i$$

$$x=\frac{-i\sqrt{383727960\sqrt{4578}-21413691138}+\sqrt{21413691138}}{9156}\approx 15,982325934-7,366910212i$$

Kvadrat tenglamalaridan foydalanish qoidalari

Ikkala tarafdan 10523 ni ayirish.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x+31478-10523=0$$

20955 olish uchun 31478 dan 10523 ni ayirish.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x+20955=0$$

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.

$$\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x+20955=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} \sqrt{4578} ni a, -\sqrt{4677521} ni b va 20955 ni c bilan almashtiring.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{\left(-\sqrt{4677521}\right)^{2}-4\sqrt{4578}\times 20955}}{2\sqrt{4578}}$$

-\sqrt{4677521} kvadratini chiqarish.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521-4\sqrt{4578}\times 20955}}{2\sqrt{4578}}$$

-4 ni \sqrt{4578} marotabaga ko'paytirish.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521+\left(-4\sqrt{4578}\right)\times 20955}}{2\sqrt{4578}}$$

-4\sqrt{4578} ni 20955 marotabaga ko'paytirish.

$$x=\frac{-\left(-\sqrt{4677521}\right)±\sqrt{4677521-83820\sqrt{4578}}}{2\sqrt{4578}}$$

4677521-83820\sqrt{4578} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-\sqrt{4677521}\right)±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}}$$

-\sqrt{4677521} ning teskarisi \sqrt{4677521} ga teng.

$$x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}}$$

x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} tenglamasini yeching, bunda ± musbat. \sqrt{4677521} ni i\sqrt{-\left(4677521-83820\sqrt{4578}\right)} ga qo'shish.

$$x=\frac{\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}}{2\sqrt{4578}}$$

\sqrt{4677521}+i\sqrt{-4677521+83820\sqrt{4578}} ni 2\sqrt{4578} ga bo'lish.

$$x=\frac{\sqrt{4578}\left(\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}\right)}{9156}$$

x=\frac{\sqrt{4677521}±i\sqrt{-\left(4677521-83820\sqrt{4578}\right)}}{2\sqrt{4578}} tenglamasini yeching, bunda ± manfiy. \sqrt{4677521} dan i\sqrt{-\left(4677521-83820\sqrt{4578}\right)} ni ayirish.

$$x=\frac{-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}}{2\sqrt{4578}}$$

\sqrt{4677521}-i\sqrt{-4677521+83820\sqrt{4578}} ni 2\sqrt{4578} ga bo'lish.

$$x=\frac{\sqrt{4578}\left(-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}\right)}{9156}$$

Tenglama yechildi.

$$x=\frac{\sqrt{4578}\left(\sqrt{4677521}+i\sqrt{83820\sqrt{4578}-4677521}\right)}{9156} x=\frac{\sqrt{4578}\left(-i\sqrt{83820\sqrt{4578}-4677521}+\sqrt{4677521}\right)}{9156}$$

Kvadratni yakunlash bo'yicha qadamlar

Ikkala tarafdan 31478 ni ayirish.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x=10523-31478$$

-20955 olish uchun 10523 dan 31478 ni ayirish.

$$\sqrt{4578}x^{2}-\sqrt{4677521}x=-20955$$

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

$$\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x=-20955$$

Ikki tarafini \sqrt{4578} ga bo‘ling.

$$\frac{\sqrt{4578}x^{2}+\left(-\sqrt{4677521}\right)x}{\sqrt{4578}}=-\frac{20955}{\sqrt{4578}}$$

\sqrt{4578} ga bo'lish \sqrt{4578} ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{\sqrt{4677521}}{\sqrt{4578}}\right)x=-\frac{20955}{\sqrt{4578}}$$

-\sqrt{4677521} ni \sqrt{4578} ga bo'lish.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x=-\frac{20955}{\sqrt{4578}}$$

-20955 ni \sqrt{4578} ga bo'lish.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x=-\frac{6985\sqrt{4578}}{1526}$$

-\frac{\sqrt{21413691138}}{4578} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{\sqrt{21413691138}}{9156} olish uchun. Keyin, -\frac{\sqrt{21413691138}}{9156} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\left(-\frac{\sqrt{21413691138}}{9156}\right)^{2}=-\frac{6985\sqrt{4578}}{1526}+\left(-\frac{\sqrt{21413691138}}{9156}\right)^{2}$$

-\frac{\sqrt{21413691138}}{9156} kvadratini chiqarish.

$$x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\frac{4677521}{18312}=-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}$$

x^{2}+\left(-\frac{\sqrt{21413691138}}{4578}\right)x+\frac{4677521}{18312} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{\sqrt{21413691138}}{9156}\right)^{2}=-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{\sqrt{21413691138}}{9156}\right)^{2}}=\sqrt{-\frac{6985\sqrt{4578}}{1526}+\frac{4677521}{18312}}$$

Qisqartirish.

$$x-\frac{\sqrt{21413691138}}{9156}=\frac{i\sqrt{383727960\sqrt{4578}-21413691138}}{9156} x-\frac{\sqrt{21413691138}}{9156}=-\frac{i\sqrt{383727960\sqrt{4578}-21413691138}}{9156}$$

\frac{\sqrt{21413691138}}{9156} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{21413691138}+i\sqrt{383727960\sqrt{4578}-21413691138}}{9156} x=\frac{-i\sqrt{383727960\sqrt{4578}-21413691138}+\sqrt{21413691138}}{9156}$$