-8/9=4(x-2)(x-3) eching

$$- \frac { 8 } { 9 } = 4 ( x - 2 ) ( x - 3 )$$

x uchun yechish

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

$$x = \frac{7}{3} = 2\frac{1}{3} \approx 2,333333333$$

Kvadrat tenglamalaridan foydalanish qoidalari

4 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-\frac{8}{9}=\left(4x-8\right)\left(x-3\right)$$

4x-8 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-\frac{8}{9}=4x^{2}-20x+24$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$4x^{2}-20x+24=-\frac{8}{9}$$

\frac{8}{9} ni ikki tarafga qo’shing.

$$4x^{2}-20x+24+\frac{8}{9}=0$$

\frac{224}{9} olish uchun 24 va \frac{8}{9}'ni qo'shing.

$$4x^{2}-20x+\frac{224}{9}=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} 4 ni a, -20 ni b va \frac{224}{9} ni c bilan almashtiring.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 4\times \frac{224}{9}}}{2\times 4}$$

-20 kvadratini chiqarish.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 4\times \frac{224}{9}}}{2\times 4}$$

-4 ni 4 marotabaga ko'paytirish.

$$x=\frac{-\left(-20\right)±\sqrt{400-16\times \frac{224}{9}}}{2\times 4}$$

-16 ni \frac{224}{9} marotabaga ko'paytirish.

$$x=\frac{-\left(-20\right)±\sqrt{400-\frac{3584}{9}}}{2\times 4}$$

400 ni -\frac{3584}{9} ga qo'shish.

$$x=\frac{-\left(-20\right)±\sqrt{\frac{16}{9}}}{2\times 4}$$

\frac{16}{9} ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-20\right)±\frac{4}{3}}{2\times 4}$$

-20 ning teskarisi 20 ga teng.

$$x=\frac{20±\frac{4}{3}}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$x=\frac{20±\frac{4}{3}}{8}$$

x=\frac{20±\frac{4}{3}}{8} tenglamasini yeching, bunda ± musbat. 20 ni \frac{4}{3} ga qo'shish.

$$x=\frac{\frac{64}{3}}{8}$$

\frac{64}{3} ni 8 ga bo'lish.

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

x=\frac{20±\frac{4}{3}}{8} tenglamasini yeching, bunda ± manfiy. 20 dan \frac{4}{3} ni ayirish.

$$x=\frac{\frac{56}{3}}{8}$$

\frac{56}{3} ni 8 ga bo'lish.

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

Tenglama yechildi.

$$x=\frac{8}{3} x=\frac{7}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

4 ga x-2 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$-\frac{8}{9}=\left(4x-8\right)\left(x-3\right)$$

4x-8 ga x-3 ni ko‘paytirish orqali distributiv xususiyatdan foydalaning va ifoda sifatida birlashtiring.

$$-\frac{8}{9}=4x^{2}-20x+24$$

Tomonlarni almashtirib, barcha oʻzgaruvchi shartlar chap tomonga oʻtkazing.

$$4x^{2}-20x+24=-\frac{8}{9}$$

Ikkala tarafdan 24 ni ayirish.

$$4x^{2}-20x=-\frac{8}{9}-24$$

-\frac{224}{9} olish uchun -\frac{8}{9} dan 24 ni ayirish.

$$4x^{2}-20x=-\frac{224}{9}$$

Ikki tarafini 4 ga bo‘ling.

$$\frac{4x^{2}-20x}{4}=-\frac{\frac{224}{9}}{4}$$

4 ga bo'lish 4 ga ko'paytirishni bekor qiladi.

$$x^{2}+\left(-\frac{20}{4}\right)x=-\frac{\frac{224}{9}}{4}$$

-20 ni 4 ga bo'lish.

$$x^{2}-5x=-\frac{\frac{224}{9}}{4}$$

-\frac{224}{9} ni 4 ga bo'lish.

$$x^{2}-5x=-\frac{56}{9}$$

-5 ni bo‘lish, x shartining koeffitsienti, 2 ga -\frac{5}{2} olish uchun. Keyin, -\frac{5}{2} ning kvadratini tenglamaning ikkala tarafiga qo‘shing. Ushbu qadam tenglamaning chap qismini mukammal kvadrat sifatida hosil qiladi.

$$x^{2}-5x+\left(-\frac{5}{2}\right)^{2}=-\frac{56}{9}+\left(-\frac{5}{2}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{5}{2} kvadratini chiqarish.

$$x^{2}-5x+\frac{25}{4}=-\frac{56}{9}+\frac{25}{4}$$

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{56}{9} ni \frac{25}{4} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-5x+\frac{25}{4}=\frac{1}{36}$$

x^{2}-5x+\frac{25}{4} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{5}{2}\right)^{2}=\frac{1}{36}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Qisqartirish.

$$x-\frac{5}{2}=\frac{1}{6} x-\frac{5}{2}=-\frac{1}{6}$$

\frac{5}{2} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{8}{3} x=\frac{7}{3}$$