(4x-1)^2=(x-1)(x+1) eching

$$( 4 x - 1 ) ^ { 2 } = ( x - 1 ) ( x + 1 )$$

x uchun yechish (complex solution)

$$x=\frac{4+\sqrt{14}i}{15}\approx 0,266666667+0,249443826i$$

$$x=\frac{-\sqrt{14}i+4}{15}\approx 0,266666667-0,249443826i$$

Kvadrat tenglamalaridan foydalanish qoidalari

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-1\right)^{2} kengaytirilishi uchun ishlating.

$$16x^{2}-8x+1=\left(x-1\right)\left(x+1\right)$$

Hisoblang: \left(x-1\right)\left(x+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

$$16x^{2}-8x+1=x^{2}-1$$

Ikkala tarafdan x^{2} ni ayirish.

$$16x^{2}-8x+1-x^{2}=-1$$

15x^{2} ni olish uchun 16x^{2} va -x^{2} ni birlashtirish.

$$15x^{2}-8x+1=-1$$

1 ni ikki tarafga qo’shing.

$$15x^{2}-8x+1+1=0$$

2 olish uchun 1 va 1'ni qo'shing.

$$15x^{2}-8x+2=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 15 ni a, -8 ni b va 2 ni c bilan almashtiring.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15\times 2}}{2\times 15}$$

-8 kvadratini chiqarish.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 15\times 2}}{2\times 15}$$

-4 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-60\times 2}}{2\times 15}$$

-60 ni 2 marotabaga ko'paytirish.

$$x=\frac{-\left(-8\right)±\sqrt{64-120}}{2\times 15}$$

64 ni -120 ga qo'shish.

$$x=\frac{-\left(-8\right)±\sqrt{-56}}{2\times 15}$$

-56 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-8\right)±2\sqrt{14}i}{2\times 15}$$

-8 ning teskarisi 8 ga teng.

$$x=\frac{8±2\sqrt{14}i}{2\times 15}$$

2 ni 15 marotabaga ko'paytirish.

$$x=\frac{8±2\sqrt{14}i}{30}$$

x=\frac{8±2\sqrt{14}i}{30} tenglamasini yeching, bunda ± musbat. 8 ni 2i\sqrt{14} ga qo'shish.

$$x=\frac{8+2\sqrt{14}i}{30}$$

8+2i\sqrt{14} ni 30 ga bo'lish.

$$x=\frac{4+\sqrt{14}i}{15}$$

x=\frac{8±2\sqrt{14}i}{30} tenglamasini yeching, bunda ± manfiy. 8 dan 2i\sqrt{14} ni ayirish.

$$x=\frac{-2\sqrt{14}i+8}{30}$$

8-2i\sqrt{14} ni 30 ga bo'lish.

$$x=\frac{-\sqrt{14}i+4}{15}$$

Tenglama yechildi.

$$x=\frac{4+\sqrt{14}i}{15} x=\frac{-\sqrt{14}i+4}{15}$$

Kvadratni yakunlash bo'yicha qadamlar

\left(a-b\right)^{2}=a^{2}-2ab+b^{2} binom teoremasini \left(4x-1\right)^{2} kengaytirilishi uchun ishlating.

$$16x^{2}-8x+1=\left(x-1\right)\left(x+1\right)$$

Hisoblang: \left(x-1\right)\left(x+1\right). Ko‘paytirish qoida yordamida turli kvadratlarga aylantirilishi mumkin: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. 1 kvadratini chiqarish.

$$16x^{2}-8x+1=x^{2}-1$$

Ikkala tarafdan x^{2} ni ayirish.

$$16x^{2}-8x+1-x^{2}=-1$$

15x^{2} ni olish uchun 16x^{2} va -x^{2} ni birlashtirish.

$$15x^{2}-8x+1=-1$$

Ikkala tarafdan 1 ni ayirish.

$$15x^{2}-8x=-1-1$$

-2 olish uchun -1 dan 1 ni ayirish.

$$15x^{2}-8x=-2$$

Ikki tarafini 15 ga bo‘ling.

$$\frac{15x^{2}-8x}{15}=-\frac{2}{15}$$

15 ga bo'lish 15 ga ko'paytirishni bekor qiladi.

$$x^{2}-\frac{8}{15}x=-\frac{2}{15}$$

-\frac{8}{15} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{4}{15} olish uchun. Keyin, -\frac{4}{15} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{8}{15}x+\left(-\frac{4}{15}\right)^{2}=-\frac{2}{15}+\left(-\frac{4}{15}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{4}{15} kvadratini chiqarish.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=-\frac{2}{15}+\frac{16}{225}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2}{15} ni \frac{16}{225} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=-\frac{14}{225}$$

x^{2}-\frac{8}{15}x+\frac{16}{225} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{4}{15}\right)^{2}=-\frac{14}{225}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{4}{15}\right)^{2}}=\sqrt{-\frac{14}{225}}$$

Qisqartirish.

$$x-\frac{4}{15}=\frac{\sqrt{14}i}{15} x-\frac{4}{15}=-\frac{\sqrt{14}i}{15}$$

\frac{4}{15} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{4+\sqrt{14}i}{15} x=\frac{-\sqrt{14}i+4}{15}$$