=8x^2-22x+15 eching

$$= 8 x ^ { 2 } - 22 x + 15$$

Omil

$$\left(2x-3\right)\left(4x-5\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 8x^{2}+ax+bx+15 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-22 ab=8\times 15=120$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 120-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-120 -2,-60 -3,-40 -4,-30 -5,-24 -6,-20 -8,-15 -10,-12$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-120=-121 -2-60=-62 -3-40=-43 -4-30=-34 -5-24=-29 -6-20=-26 -8-15=-23 -10-12=-22$$

Yechim – -22 yigʻindisini beruvchi juftlik.

$$a=-12 b=-10$$

8x^{2}-22x+15 ni \left(8x^{2}-12x\right)+\left(-10x+15\right) sifatida qaytadan yozish.

$$\left(8x^{2}-12x\right)+\left(-10x+15\right)$$

Birinchi guruhda 4x ni va ikkinchi guruhda -5 ni faktordan chiqaring.

$$4x\left(2x-3\right)-5\left(2x-3\right)$$

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

$$\left(2x-3\right)\left(4x-5\right)$$

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.

$$8x^{2}-22x+15=0$$

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.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\times 15}}{2\times 8}$$

-22 kvadratini chiqarish.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 8\times 15}}{2\times 8}$$

-4 ni 8 marotabaga ko'paytirish.

$$x=\frac{-\left(-22\right)±\sqrt{484-32\times 15}}{2\times 8}$$

-32 ni 15 marotabaga ko'paytirish.

$$x=\frac{-\left(-22\right)±\sqrt{484-480}}{2\times 8}$$

484 ni -480 ga qo'shish.

$$x=\frac{-\left(-22\right)±\sqrt{4}}{2\times 8}$$

4 ning kvadrat ildizini chiqarish.

$$x=\frac{-\left(-22\right)±2}{2\times 8}$$

-22 ning teskarisi 22 ga teng.

$$x=\frac{22±2}{2\times 8}$$

2 ni 8 marotabaga ko'paytirish.

$$x=\frac{22±2}{16}$$

x=\frac{22±2}{16} tenglamasini yeching, bunda ± musbat. 22 ni 2 ga qo'shish.

$$x=\frac{24}{16}$$

\frac{24}{16} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

x=\frac{22±2}{16} tenglamasini yeching, bunda ± manfiy. 22 dan 2 ni ayirish.

$$x=\frac{20}{16}$$

\frac{20}{16} ulushini 4 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$x=\frac{5}{4}$$

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{5}{4} ga bo‘ling.

$$8x^{2}-22x+15=8\left(x-\frac{3}{2}\right)\left(x-\frac{5}{4}\right)$$

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

$$8x^{2}-22x+15=8\times \frac{2x-3}{2}\left(x-\frac{5}{4}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{5}{4} ni x dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$8x^{2}-22x+15=8\times \frac{2x-3}{2}\times \frac{4x-5}{4}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{2x-3}{2} ni \frac{4x-5}{4} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$8x^{2}-22x+15=8\times \frac{\left(2x-3\right)\left(4x-5\right)}{2\times 4}$$

2 ni 4 marotabaga ko'paytirish.

$$8x^{2}-22x+15=8\times \frac{\left(2x-3\right)\left(4x-5\right)}{8}$$

8 va 8 ichida eng katta umumiy 8 faktorini bekor qiling.

$$8x^{2}-22x+15=\left(2x-3\right)\left(4x-5\right)$$

Baholash

$$\left(2x-3\right)\left(4x-5\right)$$