-110=37587x+-491x^2 eching

$$-110 = 37587x+-491 { x }^{ 2 }$$

x uchun yechish

$$x = \frac{\sqrt{1412998609} + 37587}{982} \approx 76,554861259$$

$$x=\frac{37587-\sqrt{1412998609}}{982}\approx -0,002926432$$

Kvadrat tenglamalaridan foydalanish qoidalari

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$37587x-491x^{2}=-110$$

110 ni ikki tarafga qo’shing.

$$37587x-491x^{2}+110=0$$

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.

$$-491x^{2}+37587x+110=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -491 ni a, 37587 ni b va 110 ni c bilan almashtiring.

$$x=\frac{-37587±\sqrt{37587^{2}-4\left(-491\right)\times 110}}{2\left(-491\right)}$$

37587 kvadratini chiqarish.

$$x=\frac{-37587±\sqrt{1412782569-4\left(-491\right)\times 110}}{2\left(-491\right)}$$

-4 ni -491 marotabaga ko'paytirish.

$$x=\frac{-37587±\sqrt{1412782569+1964\times 110}}{2\left(-491\right)}$$

1964 ni 110 marotabaga ko'paytirish.

$$x=\frac{-37587±\sqrt{1412782569+216040}}{2\left(-491\right)}$$

1412782569 ni 216040 ga qo'shish.

$$x=\frac{-37587±\sqrt{1412998609}}{2\left(-491\right)}$$

2 ni -491 marotabaga ko'paytirish.

$$x=\frac{-37587±\sqrt{1412998609}}{-982}$$

x=\frac{-37587±\sqrt{1412998609}}{-982} tenglamasini yeching, bunda ± musbat. -37587 ni \sqrt{1412998609} ga qo'shish.

$$x=\frac{\sqrt{1412998609}-37587}{-982}$$

-37587+\sqrt{1412998609} ni -982 ga bo'lish.

$$x=\frac{37587-\sqrt{1412998609}}{982}$$

x=\frac{-37587±\sqrt{1412998609}}{-982} tenglamasini yeching, bunda ± manfiy. -37587 dan \sqrt{1412998609} ni ayirish.

$$x=\frac{-\sqrt{1412998609}-37587}{-982}$$

-37587-\sqrt{1412998609} ni -982 ga bo'lish.

$$x=\frac{\sqrt{1412998609}+37587}{982}$$

Tenglama yechildi.

$$x=\frac{37587-\sqrt{1412998609}}{982} x=\frac{\sqrt{1412998609}+37587}{982}$$

Kvadratni yakunlash bo'yicha qadamlar

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$37587x-491x^{2}=-110$$

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

$$-491x^{2}+37587x=-110$$

Ikki tarafini -491 ga bo‘ling.

$$\frac{-491x^{2}+37587x}{-491}=-\frac{110}{-491}$$

-491 ga bo'lish -491 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{37587}{-491}x=-\frac{110}{-491}$$

37587 ni -491 ga bo'lish.

$$x^{2}-\frac{37587}{491}x=-\frac{110}{-491}$$

-110 ni -491 ga bo'lish.

$$x^{2}-\frac{37587}{491}x=\frac{110}{491}$$

-\frac{37587}{491} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{37587}{982} olish uchun. Keyin, -\frac{37587}{982} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{37587}{491}x+\left(-\frac{37587}{982}\right)^{2}=\frac{110}{491}+\left(-\frac{37587}{982}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{37587}{982} kvadratini chiqarish.

$$x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}=\frac{110}{491}+\frac{1412782569}{964324}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{110}{491} ni \frac{1412782569}{964324} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324}=\frac{1412998609}{964324}$$

x^{2}-\frac{37587}{491}x+\frac{1412782569}{964324} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{37587}{982}\right)^{2}=\frac{1412998609}{964324}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{37587}{982}\right)^{2}}=\sqrt{\frac{1412998609}{964324}}$$

Qisqartirish.

$$x-\frac{37587}{982}=\frac{\sqrt{1412998609}}{982} x-\frac{37587}{982}=-\frac{\sqrt{1412998609}}{982}$$

\frac{37587}{982} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{1412998609}+37587}{982} x=\frac{37587-\sqrt{1412998609}}{982}$$