5x^2+18x+1=0 eching

$$5 x ^ { 2 } + 18 x + 1 = 0$$

x uchun yechish

$$x=\frac{2\sqrt{19}-9}{5}\approx -0,056440423$$

$$x=\frac{-2\sqrt{19}-9}{5}\approx -3,543559577$$

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.

$$5x^{2}+18x+1=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 5 ni a, 18 ni b va 1 ni c bilan almashtiring.

$$x=\frac{-18±\sqrt{18^{2}-4\times 5}}{2\times 5}$$

18 kvadratini chiqarish.

$$x=\frac{-18±\sqrt{324-4\times 5}}{2\times 5}$$

-4 ni 5 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324-20}}{2\times 5}$$

324 ni -20 ga qo'shish.

$$x=\frac{-18±\sqrt{304}}{2\times 5}$$

304 ning kvadrat ildizini chiqarish.

$$x=\frac{-18±4\sqrt{19}}{2\times 5}$$

2 ni 5 marotabaga ko'paytirish.

$$x=\frac{-18±4\sqrt{19}}{10}$$

x=\frac{-18±4\sqrt{19}}{10} tenglamasini yeching, bunda ± musbat. -18 ni 4\sqrt{19} ga qo'shish.

$$x=\frac{4\sqrt{19}-18}{10}$$

-18+4\sqrt{19} ni 10 ga bo'lish.

$$x=\frac{2\sqrt{19}-9}{5}$$

x=\frac{-18±4\sqrt{19}}{10} tenglamasini yeching, bunda ± manfiy. -18 dan 4\sqrt{19} ni ayirish.

$$x=\frac{-4\sqrt{19}-18}{10}$$

-18-4\sqrt{19} ni 10 ga bo'lish.

$$x=\frac{-2\sqrt{19}-9}{5}$$

Tenglama yechildi.

$$x=\frac{2\sqrt{19}-9}{5} x=\frac{-2\sqrt{19}-9}{5}$$

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.

$$5x^{2}+18x+1=0$$

Tenglamaning ikkala tarafidan 1 ni ayirish.

$$5x^{2}+18x+1-1=-1$$

O‘zidan 1 ayirilsa 0 qoladi.

$$5x^{2}+18x=-1$$

Ikki tarafini 5 ga bo‘ling.

$$\frac{5x^{2}+18x}{5}=-\frac{1}{5}$$

5 ga bo'lish 5 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{18}{5}x=-\frac{1}{5}$$

\frac{18}{5} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{9}{5} olish uchun. Keyin, \frac{9}{5} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{18}{5}x+\left(\frac{9}{5}\right)^{2}=-\frac{1}{5}+\left(\frac{9}{5}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{9}{5} kvadratini chiqarish.

$$x^{2}+\frac{18}{5}x+\frac{81}{25}=-\frac{1}{5}+\frac{81}{25}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{1}{5} ni \frac{81}{25} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{18}{5}x+\frac{81}{25}=\frac{76}{25}$$

x^{2}+\frac{18}{5}x+\frac{81}{25} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{9}{5}\right)^{2}=\frac{76}{25}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{9}{5}\right)^{2}}=\sqrt{\frac{76}{25}}$$

Qisqartirish.

$$x+\frac{9}{5}=\frac{2\sqrt{19}}{5} x+\frac{9}{5}=-\frac{2\sqrt{19}}{5}$$

Tenglamaning ikkala tarafidan \frac{9}{5} ni ayirish.

$$x=\frac{2\sqrt{19}-9}{5} x=\frac{-2\sqrt{19}-9}{5}$$