(x-2)(3x-5)=1 eching

$$(x-2)(3x-5)=1$$

x uchun yechish

$$x = \frac{\sqrt{13} + 11}{6} \approx 2,434258546$$

$$x = \frac{11 - \sqrt{13}}{6} \approx 1,232408121$$

Kvadrat tenglamalaridan foydalanish qoidalari

x-2 ga 3x-5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-11x+10=1$$

Ikkala tarafdan 1 ni ayirish.

$$3x^{2}-11x+10-1=0$$

9 olish uchun 10 dan 1 ni ayirish.

$$3x^{2}-11x+9=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 3 ni a, -11 ni b va 9 ni c bilan almashtiring.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 3\times 9}}{2\times 3}$$

-11 kvadratini chiqarish.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 3\times 9}}{2\times 3}$$

-4 ni 3 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-12\times 9}}{2\times 3}$$

-12 ni 9 marotabaga ko'paytirish.

$$x=\frac{-\left(-11\right)±\sqrt{121-108}}{2\times 3}$$

121 ni -108 ga qo'shish.

$$x=\frac{-\left(-11\right)±\sqrt{13}}{2\times 3}$$

-11 ning teskarisi 11 ga teng.

$$x=\frac{11±\sqrt{13}}{2\times 3}$$

2 ni 3 marotabaga ko'paytirish.

$$x=\frac{11±\sqrt{13}}{6}$$

x=\frac{11±\sqrt{13}}{6} tenglamasini yeching, bunda ± musbat. 11 ni \sqrt{13} ga qo'shish.

$$x=\frac{\sqrt{13}+11}{6}$$

x=\frac{11±\sqrt{13}}{6} tenglamasini yeching, bunda ± manfiy. 11 dan \sqrt{13} ni ayirish.

$$x=\frac{11-\sqrt{13}}{6}$$

Tenglama yechildi.

$$x=\frac{\sqrt{13}+11}{6} x=\frac{11-\sqrt{13}}{6}$$

Kvadratni yakunlash bo'yicha qadamlar

x-2 ga 3x-5 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$3x^{2}-11x+10=1$$

Ikkala tarafdan 10 ni ayirish.

$$3x^{2}-11x=1-10$$

-9 olish uchun 1 dan 10 ni ayirish.

$$3x^{2}-11x=-9$$

Ikki tarafini 3 ga bo‘ling.

$$\frac{3x^{2}-11x}{3}=-\frac{9}{3}$$

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

$$x^{2}-\frac{11}{3}x=-\frac{9}{3}$$

-9 ni 3 ga bo'lish.

$$x^{2}-\frac{11}{3}x=-3$$

-\frac{11}{3} ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{11}{6} olish uchun. Keyin, -\frac{11}{6} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-\frac{11}{3}x+\left(-\frac{11}{6}\right)^{2}=-3+\left(-\frac{11}{6}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{11}{6} kvadratini chiqarish.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=-3+\frac{121}{36}$$

-3 ni \frac{121}{36} ga qo'shish.

$$x^{2}-\frac{11}{3}x+\frac{121}{36}=\frac{13}{36}$$

x^{2}-\frac{11}{3}x+\frac{121}{36} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{11}{6}\right)^{2}=\frac{13}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{11}{6}\right)^{2}}=\sqrt{\frac{13}{36}}$$

Qisqartirish.

$$x-\frac{11}{6}=\frac{\sqrt{13}}{6} x-\frac{11}{6}=-\frac{\sqrt{13}}{6}$$

\frac{11}{6} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{\sqrt{13}+11}{6} x=\frac{11-\sqrt{13}}{6}$$