1000x^2+6125x+125=0 eching

$$1000 { x }^{ 2 } +6125x+125=0$$

x uchun yechish

$$x=\frac{\sqrt{2369}-49}{16}\approx -0,020476619$$

$$x=\frac{-\sqrt{2369}-49}{16}\approx -6,104523381$$

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.

$$1000x^{2}+6125x+125=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 1000 ni a, 6125 ni b va 125 ni c bilan almashtiring.

$$x=\frac{-6125±\sqrt{6125^{2}-4\times 1000\times 125}}{2\times 1000}$$

6125 kvadratini chiqarish.

$$x=\frac{-6125±\sqrt{37515625-4\times 1000\times 125}}{2\times 1000}$$

-4 ni 1000 marotabaga ko'paytirish.

$$x=\frac{-6125±\sqrt{37515625-4000\times 125}}{2\times 1000}$$

-4000 ni 125 marotabaga ko'paytirish.

$$x=\frac{-6125±\sqrt{37515625-500000}}{2\times 1000}$$

37515625 ni -500000 ga qo'shish.

$$x=\frac{-6125±\sqrt{37015625}}{2\times 1000}$$

37015625 ning kvadrat ildizini chiqarish.

$$x=\frac{-6125±125\sqrt{2369}}{2\times 1000}$$

2 ni 1000 marotabaga ko'paytirish.

$$x=\frac{-6125±125\sqrt{2369}}{2000}$$

x=\frac{-6125±125\sqrt{2369}}{2000} tenglamasini yeching, bunda ± musbat. -6125 ni 125\sqrt{2369} ga qo'shish.

$$x=\frac{125\sqrt{2369}-6125}{2000}$$

-6125+125\sqrt{2369} ni 2000 ga bo'lish.

$$x=\frac{\sqrt{2369}-49}{16}$$

x=\frac{-6125±125\sqrt{2369}}{2000} tenglamasini yeching, bunda ± manfiy. -6125 dan 125\sqrt{2369} ni ayirish.

$$x=\frac{-125\sqrt{2369}-6125}{2000}$$

-6125-125\sqrt{2369} ni 2000 ga bo'lish.

$$x=\frac{-\sqrt{2369}-49}{16}$$

Tenglama yechildi.

$$x=\frac{\sqrt{2369}-49}{16} x=\frac{-\sqrt{2369}-49}{16}$$

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.

$$1000x^{2}+6125x+125=0$$

Tenglamaning ikkala tarafidan 125 ni ayirish.

$$1000x^{2}+6125x+125-125=-125$$

O‘zidan 125 ayirilsa 0 qoladi.

$$1000x^{2}+6125x=-125$$

Ikki tarafini 1000 ga bo‘ling.

$$\frac{1000x^{2}+6125x}{1000}=-\frac{125}{1000}$$

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

$$x^{2}+\frac{6125}{1000}x=-\frac{125}{1000}$$

\frac{6125}{1000} ulushini 125 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{49}{8}x=-\frac{125}{1000}$$

\frac{-125}{1000} ulushini 125 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{49}{8}x=-\frac{1}{8}$$

\frac{49}{8} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{49}{16} olish uchun. Keyin, \frac{49}{16} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{49}{8}x+\left(\frac{49}{16}\right)^{2}=-\frac{1}{8}+\left(\frac{49}{16}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{49}{16} kvadratini chiqarish.

$$x^{2}+\frac{49}{8}x+\frac{2401}{256}=-\frac{1}{8}+\frac{2401}{256}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{8} ni \frac{2401}{256} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{49}{8}x+\frac{2401}{256}=\frac{2369}{256}$$

x^{2}+\frac{49}{8}x+\frac{2401}{256} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{49}{16}\right)^{2}=\frac{2369}{256}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{49}{16}\right)^{2}}=\sqrt{\frac{2369}{256}}$$

Qisqartirish.

$$x+\frac{49}{16}=\frac{\sqrt{2369}}{16} x+\frac{49}{16}=-\frac{\sqrt{2369}}{16}$$

Tenglamaning ikkala tarafidan \frac{49}{16} ni ayirish.

$$x=\frac{\sqrt{2369}-49}{16} x=\frac{-\sqrt{2369}-49}{16}$$