(x-1)(3x+1)=15 eching

$$( x - 1 ) ( 3 x + 1 ) = 15$$

x uchun yechish

$$x=-2$$

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2,666666667$$

Kvadrat tenglamalaridan foydalanish qoidalari

x-1 ga 3x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-2x-1=15$$

Ikkala tarafdan 15 ni ayirish.

$$3x^{2}-2x-1-15=0$$

-16 olish uchun -1 dan 15 ni ayirish.

$$3x^{2}-2x-16=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -2 ni b va -16 ni c bilan almashtiring.

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

-2 kvadratini chiqarish.

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

-4 ni 3 marotabaga ko'paytirish.

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

-12 ni -16 marotabaga ko'paytirish.

$$x=\frac{-\left(-2\right)±\sqrt{4+192}}{2\times 3}$$

4 ni 192 ga qo'shish.

$$x=\frac{-\left(-2\right)±\sqrt{196}}{2\times 3}$$

196 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-2\right)±14}{2\times 3}$$

-2 ning teskarisi 2 ga teng.

$$x=\frac{2±14}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{2±14}{6}$$

x=\frac{2±14}{6} tenglamasini yeching, bunda ± musbat. 2 ni 14 ga qo'shish.

$$x=\frac{16}{6}$$

\frac{16}{6} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{2±14}{6} tenglamasini yeching, bunda ± manfiy. 2 dan 14 ni ayirish.

$$x=-\frac{12}{6}$$

-12 ni 6 ga bo'lish.

$$x=-2$$

Tenglama yechildi.

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

Kvadratni yakunlash bo'yicha qadamlar

x-1 ga 3x+1 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-2x-1=15$$

1 ni ikki tarafga qo’shing.

$$3x^{2}-2x=15+1$$

16 olish uchun 15 va 1'ni qo'shing.

$$3x^{2}-2x=16$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-2x}{3}=\frac{16}{3}$$

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

$$x^{2}-\frac{2}{3}x=\frac{16}{3}$$

-\frac{2}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{1}{3} olish uchun. Keyin, -\frac{1}{3} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{2}{3}x+\left(-\frac{1}{3}\right)^{2}=\frac{16}{3}+\left(-\frac{1}{3}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{1}{3} kvadratini chiqarish.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{16}{3}+\frac{1}{9}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali \frac{16}{3} ni \frac{1}{9} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{2}{3}x+\frac{1}{9}=\frac{49}{9}$$

x^{2}-\frac{2}{3}x+\frac{1}{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.

$$\left(x-\frac{1}{3}\right)^{2}=\frac{49}{9}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{1}{3}\right)^{2}}=\sqrt{\frac{49}{9}}$$

Qisqartirish.

$$x-\frac{1}{3}=\frac{7}{3} x-\frac{1}{3}=-\frac{7}{3}$$

\frac{1}{3} ni tenglamaning ikkala tarafiga qo'shish.

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