-18x^2=-27x+4 eching

$$- 18 x ^ { 2 } = - 27 x + 4$$

x uchun yechish

$$x=\frac{1}{6}\approx 0,166666667$$

$$x = \frac{4}{3} = 1\frac{1}{3} \approx 1,333333333$$

Guruhlash orqali faktorlash qadamlari

27x ni ikki tarafga qo’shing.

$$-18x^{2}+27x=4$$

Ikkala tarafdan 4 ni ayirish.

$$-18x^{2}+27x-4=0$$

Tenglamani yechish uchun guruhlash orqali chap qoʻl tomonni faktorlang. Avvalo, chap qoʻl tomon -18x^{2}+ax+bx-4 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=27 ab=-18\left(-4\right)=72$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b musbat boʻlganda, a va b ikkisi ham musbat. 72-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$1,72 2,36 3,24 4,18 6,12 8,9$$

Har bir juftlik yigʻindisini hisoblang.

$$1+72=73 2+36=38 3+24=27 4+18=22 6+12=18 8+9=17$$

Yechim – 27 yigʻindisini beruvchi juftlik.

$$a=24 b=3$$

-18x^{2}+27x-4 ni \left(-18x^{2}+24x\right)+\left(3x-4\right) sifatida qaytadan yozish.

$$\left(-18x^{2}+24x\right)+\left(3x-4\right)$$

-18x^{2}+24x ichida -6x ni ajrating.

$$-6x\left(3x-4\right)+3x-4$$

Distributiv funktsiyasidan foydalangan holda 3x-4 umumiy terminini chiqaring.

$$\left(3x-4\right)\left(-6x+1\right)$$

Tenglamani yechish uchun 3x-4=0 va -6x+1=0 ni yeching.

$$x=\frac{4}{3} x=\frac{1}{6}$$

Kvadrat tenglamalaridan foydalanish qoidalari

27x ni ikki tarafga qo’shing.

$$-18x^{2}+27x=4$$

Ikkala tarafdan 4 ni ayirish.

$$-18x^{2}+27x-4=0$$

Ushbu tenglama standart shaklidadir: ax^{2}+bx+c=0. Kvadrat tenglama formulasida, \frac{-b±\sqrt{b^{2}-4ac}}{2a} -18 ni a, 27 ni b va -4 ni c bilan almashtiring.

$$x=\frac{-27±\sqrt{27^{2}-4\left(-18\right)\left(-4\right)}}{2\left(-18\right)}$$

27 kvadratini chiqarish.

$$x=\frac{-27±\sqrt{729-4\left(-18\right)\left(-4\right)}}{2\left(-18\right)}$$

-4 ni -18 marotabaga ko'paytirish.

$$x=\frac{-27±\sqrt{729+72\left(-4\right)}}{2\left(-18\right)}$$

72 ni -4 marotabaga ko'paytirish.

$$x=\frac{-27±\sqrt{729-288}}{2\left(-18\right)}$$

729 ni -288 ga qo'shish.

$$x=\frac{-27±\sqrt{441}}{2\left(-18\right)}$$

441 ning kvadrat ildizini chiqarish.

$$x=\frac{-27±21}{2\left(-18\right)}$$

2 ni -18 marotabaga ko'paytirish.

$$x=\frac{-27±21}{-36}$$

x=\frac{-27±21}{-36} tenglamasini yeching, bunda ± musbat. -27 ni 21 ga qo'shish.

$$x=-\frac{6}{-36}$$

\frac{-6}{-36} ulushini 6 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{1}{6}$$

x=\frac{-27±21}{-36} tenglamasini yeching, bunda ± manfiy. -27 dan 21 ni ayirish.

$$x=-\frac{48}{-36}$$

\frac{-48}{-36} ulushini 12 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{4}{3}$$

Tenglama yechildi.

$$x=\frac{1}{6} x=\frac{4}{3}$$

Kvadratni yakunlash bo'yicha qadamlar

27x ni ikki tarafga qo’shing.

$$-18x^{2}+27x=4$$

Ikki tarafini -18 ga bo‘ling.

$$\frac{-18x^{2}+27x}{-18}=\frac{4}{-18}$$

-18 ga bo'lish -18 ga ko'paytirishni bekor qiladi.

$$x^{2}+\frac{27}{-18}x=\frac{4}{-18}$$

\frac{27}{-18} ulushini 9 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x^{2}-\frac{3}{2}x=\frac{4}{-18}$$

\frac{4}{-18} ulushini 2 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

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

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=-\frac{2}{9}+\left(-\frac{3}{4}\right)^{2}$$

Kasrning ham suratini, ham maxrajini kvadratga ko'paytirib -\frac{3}{4} kvadratini chiqarish.

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

Umumiy maxrajni topib va hisoblovchini qo'shish orqali -\frac{2}{9} ni \frac{9}{16} ga qo'shing. So'ngra agar imkoni bo'lsa kasrni eng kam shartga qisqartiring.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{49}{144}$$

x^{2}-\frac{3}{2}x+\frac{9}{16} omili. Odatda, x^{2}+bx+c mukammal kvadrat bo'lsa, u doimo \left(x+\frac{b}{2}\right)^{2} omil sifatida bo'lishi mumkin.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{49}{144}$$

Tenglamaning ikkala tarafining kvadrat ildizini chiqarish.

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Qisqartirish.

$$x-\frac{3}{4}=\frac{7}{12} x-\frac{3}{4}=-\frac{7}{12}$$

\frac{3}{4} ni tenglamaning ikkala tarafiga qo'shish.

$$x=\frac{4}{3} x=\frac{1}{6}$$