54x^2-141x+90 eching

$$54 { x }^{ 2 } -141x+90$$

Omil

$$3\left(2x-3\right)\left(9x-10\right)$$

Yechim qadamlari

3 omili.

$$3\left(18x^{2}-47x+30\right)$$

Hisoblang: 18x^{2}-47x+30. Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 18x^{2}+ax+bx+30 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-47 ab=18\times 30=540$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 540-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-540 -2,-270 -3,-180 -4,-135 -5,-108 -6,-90 -9,-60 -10,-54 -12,-45 -15,-36 -18,-30 -20,-27$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-540=-541 -2-270=-272 -3-180=-183 -4-135=-139 -5-108=-113 -6-90=-96 -9-60=-69 -10-54=-64 -12-45=-57 -15-36=-51 -18-30=-48 -20-27=-47$$

Yechim – -47 yigʻindisini beruvchi juftlik.

$$a=-27 b=-20$$

18x^{2}-47x+30 ni \left(18x^{2}-27x\right)+\left(-20x+30\right) sifatida qaytadan yozish.

$$\left(18x^{2}-27x\right)+\left(-20x+30\right)$$

Birinchi guruhda 9x ni va ikkinchi guruhda -10 ni faktordan chiqaring.

$$9x\left(2x-3\right)-10\left(2x-3\right)$$

Distributiv funktsiyasidan foydalangan holda 2x-3 umumiy terminini chiqaring.

$$\left(2x-3\right)\left(9x-10\right)$$

Toʻliq ajratilgan ifodani qaytadan yozing.

$$3\left(2x-3\right)\left(9x-10\right)$$

Kvadrat tenglamalaridan foydalanish qoidalari

Kvadrat koʻp tenglama bu orqali hisoblanadi: ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), bu yerda x_{1} va x_{2} ax^{2}+bx+c=0 kvadrat tenglamaning yechimlari.

$$54x^{2}-141x+90=0$$

ax^{2}+bx+c=0 shaklidagi barcha tenglamalarni kvadrat formulasi bilan yechish mumkin: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Kvadrat formula ikki yechmni taqdim qiladi, biri ± qo'shish bo'lganda, va ikkinchisi ayiruv bo'lganda.

$$x=\frac{-\left(-141\right)±\sqrt{\left(-141\right)^{2}-4\times 54\times 90}}{2\times 54}$$

-141 kvadratini chiqarish.

$$x=\frac{-\left(-141\right)±\sqrt{19881-4\times 54\times 90}}{2\times 54}$$

-4 ni 54 marotabaga ko'paytirish.

$$x=\frac{-\left(-141\right)±\sqrt{19881-216\times 90}}{2\times 54}$$

-216 ni 90 marotabaga ko'paytirish.

$$x=\frac{-\left(-141\right)±\sqrt{19881-19440}}{2\times 54}$$

19881 ni -19440 ga qo'shish.

$$x=\frac{-\left(-141\right)±\sqrt{441}}{2\times 54}$$

441 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-141\right)±21}{2\times 54}$$

-141 ning teskarisi 141 ga teng.

$$x=\frac{141±21}{2\times 54}$$

2 ni 54 marotabaga ko'paytirish.

$$x=\frac{141±21}{108}$$

x=\frac{141±21}{108} tenglamasini yeching, bunda ± musbat. 141 ni 21 ga qo'shish.

$$x=\frac{162}{108}$$

\frac{162}{108} ulushini 54 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{141±21}{108} tenglamasini yeching, bunda ± manfiy. 141 dan 21 ni ayirish.

$$x=\frac{120}{108}$$

\frac{120}{108} ulushini 12 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{10}{9}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{3}{2} ga va x_{2} uchun \frac{10}{9} ga bo‘ling.

$$54x^{2}-141x+90=54\left(x-\frac{3}{2}\right)\left(x-\frac{10}{9}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{3}{2} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$54x^{2}-141x+90=54\times \frac{2x-3}{2}\left(x-\frac{10}{9}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{10}{9} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$54x^{2}-141x+90=54\times \frac{2x-3}{2}\times \frac{9x-10}{9}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{2x-3}{2} ni \frac{9x-10}{9} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$54x^{2}-141x+90=54\times \frac{\left(2x-3\right)\left(9x-10\right)}{2\times 9}$$

2 ni 9 marotabaga ko'paytirish.

$$54x^{2}-141x+90=54\times \frac{\left(2x-3\right)\left(9x-10\right)}{18}$$

54 va 18 ichida eng katta umumiy 18 faktorini bekor qiling.

$$54x^{2}-141x+90=3\left(2x-3\right)\left(9x-10\right)$$

Baholash

$$54x^{2}-141x+90$$