1000x^2+2x+69=0 eching

$$1000 { x }^{ 2 } +2x+69=0$$

x uchun yechish (complex solution)

$$x=\frac{-1+\sqrt{68999}i}{1000}\approx -0,001+0,262676607i$$

$$x=\frac{-\sqrt{68999}i-1}{1000}\approx -0,001-0,262676607i$$

Kvadrat tenglamalaridan foydalanish qoidalari

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.

$$1000x^{2}+2x+69=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1000 ni a, 2 ni b va 69 ni c bilan almashtiring.

$$x=\frac{-2±\sqrt{2^{2}-4\times 1000\times 69}}{2\times 1000}$$

2 kvadratini chiqarish.

$$x=\frac{-2±\sqrt{4-4\times 1000\times 69}}{2\times 1000}$$

-4 ni 1000 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-4000\times 69}}{2\times 1000}$$

-4000 ni 69 marotabaga ko'paytirish.

$$x=\frac{-2±\sqrt{4-276000}}{2\times 1000}$$

4 ni -276000 ga qo'shish.

$$x=\frac{-2±\sqrt{-275996}}{2\times 1000}$$

-275996 ning kvadrat ildizini chiqarish.

$$x=\frac{-2±2\sqrt{68999}i}{2\times 1000}$$

2 ni 1000 marotabaga ko'paytirish.

$$x=\frac{-2±2\sqrt{68999}i}{2000}$$

x=\frac{-2±2\sqrt{68999}i}{2000} tenglamasini yeching, bunda ± musbat. -2 ni 2i\sqrt{68999} ga qo'shish.

$$x=\frac{-2+2\sqrt{68999}i}{2000}$$

-2+2i\sqrt{68999} ni 2000 ga bo'lish.

$$x=\frac{-1+\sqrt{68999}i}{1000}$$

x=\frac{-2±2\sqrt{68999}i}{2000} tenglamasini yeching, bunda ± manfiy. -2 dan 2i\sqrt{68999} ni ayirish.

$$x=\frac{-2\sqrt{68999}i-2}{2000}$$

-2-2i\sqrt{68999} ni 2000 ga bo'lish.

$$x=\frac{-\sqrt{68999}i-1}{1000}$$

Tenglama yechildi.

$$x=\frac{-1+\sqrt{68999}i}{1000} x=\frac{-\sqrt{68999}i-1}{1000}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$1000x^{2}+2x+69=0$$

Tenglamaning ikkala tarafidan 69 ni ayirish.

$$1000x^{2}+2x+69-69=-69$$

O‘zidan 69 ayirilsa 0 qoladi.

$$1000x^{2}+2x=-69$$

Ikki tarafini 1000 ga bo‘ling.

$$\frac{1000x^{2}+2x}{1000}=-\frac{69}{1000}$$

1000 ga bo'lish 1000 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{2}{1000}x=-\frac{69}{1000}$$

\frac{2}{1000} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{1}{500}x=-\frac{69}{1000}$$

\frac{1}{500} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{1}{1000} olish uchun. Keyin, \frac{1}{1000} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{1}{500}x+\left(\frac{1}{1000}\right)^{2}=-\frac{69}{1000}+\left(\frac{1}{1000}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{1}{1000} kvadratini chiqarish.

$$x^{2}+\frac{1}{500}x+\frac{1}{1000000}=-\frac{69}{1000}+\frac{1}{1000000}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{69}{1000} ni \frac{1}{1000000} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{1}{500}x+\frac{1}{1000000}=-\frac{68999}{1000000}$$

x^{2}+\frac{1}{500}x+\frac{1}{1000000} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{1}{1000}\right)^{2}=-\frac{68999}{1000000}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{1}{1000}\right)^{2}}=\sqrt{-\frac{68999}{1000000}}$$

Qisqartirish.

$$x+\frac{1}{1000}=\frac{\sqrt{68999}i}{1000} x+\frac{1}{1000}=-\frac{\sqrt{68999}i}{1000}$$

Tenglamaning ikkala tarafidan \frac{1}{1000} ni ayirish.

$$x=\frac{-1+\sqrt{68999}i}{1000} x=\frac{-\sqrt{68999}i-1}{1000}$$