6x^2+18x-19=0 eching

$$6 x ^ { 2 } + 18 x - 19 = 0$$

x uchun yechish

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2}\approx 0,827373341$$

$$x=-\frac{\sqrt{195}}{6}-\frac{3}{2}\approx -3,827373341$$

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.

$$6x^{2}+18x-19=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 6 ni a, 18 ni b va -19 ni c bilan almashtiring.

$$x=\frac{-18±\sqrt{18^{2}-4\times 6\left(-19\right)}}{2\times 6}$$

18 kvadratini chiqarish.

$$x=\frac{-18±\sqrt{324-4\times 6\left(-19\right)}}{2\times 6}$$

-4 ni 6 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324-24\left(-19\right)}}{2\times 6}$$

-24 ni -19 marotabaga ko'paytirish.

$$x=\frac{-18±\sqrt{324+456}}{2\times 6}$$

324 ni 456 ga qo'shish.

$$x=\frac{-18±\sqrt{780}}{2\times 6}$$

780 ning kvadrat ildizini chiqarish.

$$x=\frac{-18±2\sqrt{195}}{2\times 6}$$

2 ni 6 marotabaga ko'paytirish.

$$x=\frac{-18±2\sqrt{195}}{12}$$

x=\frac{-18±2\sqrt{195}}{12} tenglamasini yeching, bunda ± musbat. -18 ni 2\sqrt{195} ga qo'shish.

$$x=\frac{2\sqrt{195}-18}{12}$$

-18+2\sqrt{195} ni 12 ga bo'lish.

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2}$$

x=\frac{-18±2\sqrt{195}}{12} tenglamasini yeching, bunda ± manfiy. -18 dan 2\sqrt{195} ni ayirish.

$$x=\frac{-2\sqrt{195}-18}{12}$$

-18-2\sqrt{195} ni 12 ga bo'lish.

$$x=-\frac{\sqrt{195}}{6}-\frac{3}{2}$$

Tenglama yechildi.

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2} x=-\frac{\sqrt{195}}{6}-\frac{3}{2}$$

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.

$$6x^{2}+18x-19=0$$

19 ni tenglamaning ikkala tarafiga qo'shish.

$$6x^{2}+18x-19-\left(-19\right)=-\left(-19\right)$$

O‘zidan -19 ayirilsa 0 qoladi.

$$6x^{2}+18x=-\left(-19\right)$$

0 dan -19 ni ayirish.

$$6x^{2}+18x=19$$

Ikki tarafini 6 ga bo‘ling.

$$\frac{6x^{2}+18x}{6}=\frac{19}{6}$$

6 ga bo'lish 6 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{18}{6}x=\frac{19}{6}$$

18 ni 6 ga bo'lish.

$$x^{2}+3x=\frac{19}{6}$$

3 ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{2} olish uchun. Keyin, \frac{3}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{19}{6}+\left(\frac{3}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{2} kvadratini chiqarish.

$$x^{2}+3x+\frac{9}{4}=\frac{19}{6}+\frac{9}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{19}{6} ni \frac{9}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+3x+\frac{9}{4}=\frac{65}{12}$$

x^{2}+3x+\frac{9}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{65}{12}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{65}{12}}$$

Qisqartirish.

$$x+\frac{3}{2}=\frac{\sqrt{195}}{6} x+\frac{3}{2}=-\frac{\sqrt{195}}{6}$$

Tenglamaning ikkala tarafidan \frac{3}{2} ni ayirish.

$$x=\frac{\sqrt{195}}{6}-\frac{3}{2} x=-\frac{\sqrt{195}}{6}-\frac{3}{2}$$