x^2+3x^2-x-2=0 eching

$$x ^ { 2 } + 3 x ^ { 2 } - x - 2 = 0$$

x uchun yechish

$$x=\frac{\sqrt{33}+1}{8}\approx 0,843070331$$

$$x=\frac{1-\sqrt{33}}{8}\approx -0,593070331$$

Kvadrat tenglamalaridan foydalanish qoidalari

4x^{2} ni olish uchun x^{2} va 3x^{2} ni birlashtirish.

$$4x^{2}-x-2=0$$

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

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 4\left(-2\right)}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1-16\left(-2\right)}}{2\times 4}$$

-16 ni -2 marotabaga ko'paytirish.

$$x=\frac{-\left(-1\right)±\sqrt{1+32}}{2\times 4}$$

1 ni 32 ga qo'shish.

$$x=\frac{-\left(-1\right)±\sqrt{33}}{2\times 4}$$

-1 ning teskarisi 1 ga teng.

$$x=\frac{1±\sqrt{33}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{1±\sqrt{33}}{8}$$

x=\frac{1±\sqrt{33}}{8} tenglamasini yeching, bunda ± musbat. 1 ni \sqrt{33} ga qo'shish.

$$x=\frac{\sqrt{33}+1}{8}$$

x=\frac{1±\sqrt{33}}{8} tenglamasini yeching, bunda ± manfiy. 1 dan \sqrt{33} ni ayirish.

$$x=\frac{1-\sqrt{33}}{8}$$

Tenglama yechildi.

$$x=\frac{\sqrt{33}+1}{8} x=\frac{1-\sqrt{33}}{8}$$

Kvadratni yakunlash bo'yicha qadamlar

4x^{2} ni olish uchun x^{2} va 3x^{2} ni birlashtirish.

$$4x^{2}-x-2=0$$

2 ni ikki tarafga qo’shing. Har qanday songa nolni qo‘shsangiz, o‘zi chiqadi.

$$4x^{2}-x=2$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-x}{4}=\frac{2}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{1}{4}x=\frac{2}{4}$$

\frac{2}{4} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{1}{4}x=\frac{1}{2}$$

-\frac{1}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{8} olish uchun. Keyin, -\frac{1}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{1}{4}x+\left(-\frac{1}{8}\right)^{2}=\frac{1}{2}+\left(-\frac{1}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{8} kvadratini chiqarish.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{1}{2}+\frac{1}{64}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{1}{2} ni \frac{1}{64} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{1}{4}x+\frac{1}{64}=\frac{33}{64}$$

x^{2}-\frac{1}{4}x+\frac{1}{64} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{1}{8}\right)^{2}=\frac{33}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{8}\right)^{2}}=\sqrt{\frac{33}{64}}$$

Qisqartirish.

$$x-\frac{1}{8}=\frac{\sqrt{33}}{8} x-\frac{1}{8}=-\frac{\sqrt{33}}{8}$$

\frac{1}{8} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{33}+1}{8} x=\frac{1-\sqrt{33}}{8}$$