9+x^2=(2x+1)^2 eching

$$9 + x ^ { 2 } = ( 2 x + 1 ) ^ { 2 }$$

x uchun yechish

$$x = \frac{2 \sqrt{7} - 2}{3} \approx 1,097167541$$

$$x=\frac{-2\sqrt{7}-2}{3}\approx -2,430500874$$

Kvadrat tenglamalaridan foydalanish qoidalari

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

$$9+x^{2}=4x^{2}+4x+1$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$9+x^{2}-4x^{2}=4x+1$$

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

$$9-3x^{2}=4x+1$$

Ikkala tarafdan 4x ni ayirish.

$$9-3x^{2}-4x=1$$

Ikkala tarafdan 1 ni ayirish.

$$9-3x^{2}-4x-1=0$$

8 olish uchun 9 dan 1 ni ayirish.

$$8-3x^{2}-4x=0$$

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.

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

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -3 ni a, -4 ni b va 8 ni c bilan almashtiring.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

-4 kvadratini chiqarish.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-3\right)\times 8}}{2\left(-3\right)}$$

-4 ni -3 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+12\times 8}}{2\left(-3\right)}$$

12 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-4\right)±\sqrt{16+96}}{2\left(-3\right)}$$

16 ni 96 ga qo'shish.

$$x=\frac{-\left(-4\right)±\sqrt{112}}{2\left(-3\right)}$$

112 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-4\right)±4\sqrt{7}}{2\left(-3\right)}$$

-4 ning teskarisi 4 ga teng.

$$x=\frac{4±4\sqrt{7}}{2\left(-3\right)}$$

2 ni -3 marotabaga ko'paytirish.

$$x=\frac{4±4\sqrt{7}}{-6}$$

x=\frac{4±4\sqrt{7}}{-6} tenglamasini yeching, bunda ± musbat. 4 ni 4\sqrt{7} ga qo'shish.

$$x=\frac{4\sqrt{7}+4}{-6}$$

4+4\sqrt{7} ni -6 ga bo'lish.

$$x=\frac{-2\sqrt{7}-2}{3}$$

x=\frac{4±4\sqrt{7}}{-6} tenglamasini yeching, bunda ± manfiy. 4 dan 4\sqrt{7} ni ayirish.

$$x=\frac{4-4\sqrt{7}}{-6}$$

4-4\sqrt{7} ni -6 ga bo'lish.

$$x=\frac{2\sqrt{7}-2}{3}$$

Tenglama yechildi.

$$x=\frac{-2\sqrt{7}-2}{3} x=\frac{2\sqrt{7}-2}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

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

$$9+x^{2}=4x^{2}+4x+1$$

Ikkala tarafdan 4x^{2} ni ayirish.

$$9+x^{2}-4x^{2}=4x+1$$

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

$$9-3x^{2}=4x+1$$

Ikkala tarafdan 4x ni ayirish.

$$9-3x^{2}-4x=1$$

Ikkala tarafdan 9 ni ayirish.

$$-3x^{2}-4x=1-9$$

-8 olish uchun 1 dan 9 ni ayirish.

$$-3x^{2}-4x=-8$$

Ikki tarafini -3 ga bo‘ling.

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

-3 ga bo'lish -3 ga ko'paytirishni bekor qiladi.

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

-4 ni -3 ga bo'lish.

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

-8 ni -3 ga bo'lish.

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

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

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

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

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{8}{3}+\frac{4}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{8}{3} ni \frac{4}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}+\frac{4}{3}x+\frac{4}{9}=\frac{28}{9}$$

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

$$\left(x+\frac{2}{3}\right)^{2}=\frac{28}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x+\frac{2}{3}\right)^{2}}=\sqrt{\frac{28}{9}}$$

Qisqartirish.

$$x+\frac{2}{3}=\frac{2\sqrt{7}}{3} x+\frac{2}{3}=-\frac{2\sqrt{7}}{3}$$

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

$$x=\frac{2\sqrt{7}-2}{3} x=\frac{-2\sqrt{7}-2}{3}$$