left(3x+1right)^2=x^2+1 eching

$${ \left(3x+1 \right) }^{ 2 } = { x }^{ 2 } +1$$

x uchun yechish

$$x=-\frac{3}{4}=-0,75$$

$$x=0$$

Faktorlash qadamlari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3x+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}+6x+1=x^{2}+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$9x^{2}+6x+1-x^{2}=1$$

8x^{2} ni olish uchun 9x^{2} va -x^{2} ni birlashtirish.

$$8x^{2}+6x+1=1$$

Ikkala tarafdan 1 ni ayirish.

$$8x^{2}+6x+1-1=0$$

0 olish uchun 1 dan 1 ni ayirish.

$$8x^{2}+6x=0$$

x omili.

$$x\left(8x+6\right)=0$$

Tenglamani yechish uchun x=0 va 8x+6=0 ni yeching.

$$x=0 x=-\frac{3}{4}$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3x+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}+6x+1=x^{2}+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$9x^{2}+6x+1-x^{2}=1$$

8x^{2} ni olish uchun 9x^{2} va -x^{2} ni birlashtirish.

$$8x^{2}+6x+1=1$$

Ikkala tarafdan 1 ni ayirish.

$$8x^{2}+6x+1-1=0$$

0 olish uchun 1 dan 1 ni ayirish.

$$8x^{2}+6x=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 8 ni a, 6 ni b va 0 ni c bilan almashtiring.

$$x=\frac{-6±\sqrt{6^{2}}}{2\times 8}$$

6^{2} ning kvadrat ildizini chiqarish.

$$x=\frac{-6±6}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{-6±6}{16}$$

x=\frac{-6±6}{16} tenglamasini yeching, bunda ± musbat. -6 ni 6 ga qo'shish.

$$x=\frac{0}{16}$$

0 ni 16 ga bo'lish.

$$x=0$$

x=\frac{-6±6}{16} tenglamasini yeching, bunda ± manfiy. -6 dan 6 ni ayirish.

$$x=-\frac{12}{16}$$

\frac{-12}{16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=-\frac{3}{4}$$

Tenglama yechildi.

$$x=0 x=-\frac{3}{4}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a+b\right)^{2}=a^{2}+2ab+b^{2} binom teoremasini \left(3x+1\right)^{2} kengaytirilishi uchun ishlating.

$$9x^{2}+6x+1=x^{2}+1$$

Ikkala tarafdan x^{2} ni ayirish.

$$9x^{2}+6x+1-x^{2}=1$$

8x^{2} ni olish uchun 9x^{2} va -x^{2} ni birlashtirish.

$$8x^{2}+6x+1=1$$

Ikkala tarafdan 1 ni ayirish.

$$8x^{2}+6x=1-1$$

0 olish uchun 1 dan 1 ni ayirish.

$$8x^{2}+6x=0$$

Ikki tarafini 8 ga bo‘ling.

$$\frac{8x^{2}+6x}{8}=\frac{0}{8}$$

8 ga bo'lish 8 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{6}{8}x=\frac{0}{8}$$

\frac{6}{8} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}+\frac{3}{4}x=\frac{0}{8}$$

0 ni 8 ga bo'lish.

$$x^{2}+\frac{3}{4}x=0$$

\frac{3}{4} ni bo‘lish, x shartining koeffitsienti, 2 ga \frac{3}{8} olish uchun. Keyin, \frac{3}{8} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}+\frac{3}{4}x+\left(\frac{3}{8}\right)^{2}=\left(\frac{3}{8}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib \frac{3}{8} kvadratini chiqarish.

$$x^{2}+\frac{3}{4}x+\frac{9}{64}=\frac{9}{64}$$

x^{2}+\frac{3}{4}x+\frac{9}{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.

$$\left(x+\frac{3}{8}\right)^{2}=\frac{9}{64}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{3}{8}\right)^{2}}=\sqrt{\frac{9}{64}}$$

Qisqartirish.

$$x+\frac{3}{8}=\frac{3}{8} x+\frac{3}{8}=-\frac{3}{8}$$

Tenglamaning ikkala tarafidan \frac{3}{8} ni ayirish.

$$x=0 x=-\frac{3}{4}$$