52z^2-43z+3 eching

$$52 z ^ { 2 } - 43 z + 3$$

Omil

$$\left(4z-3\right)\left(13z-1\right)$$

Yechim qadamlari

Ifodani guruhlash orqali faktorlang. Avvalo, ifoda 52z^{2}+az+bz+3 sifatida qayta yozilishi kerak. a va b ni topish uchun yechiladigan tizimni sozlang.

$$a+b=-43 ab=52\times 3=156$$

ab musbat boʻlganda, a va b da bir xil belgi bor. a+b manfiy boʻlganda, a va b ikkisi ham manfiy. 156-mahsulotni beruvchi bunday butun juftliklarni roʻyxat qiling.

$$-1,-156 -2,-78 -3,-52 -4,-39 -6,-26 -12,-13$$

Har bir juftlik yigʻindisini hisoblang.

$$-1-156=-157 -2-78=-80 -3-52=-55 -4-39=-43 -6-26=-32 -12-13=-25$$

Yechim – -43 yigʻindisini beruvchi juftlik.

$$a=-39 b=-4$$

52z^{2}-43z+3 ni \left(52z^{2}-39z\right)+\left(-4z+3\right) sifatida qaytadan yozish.

$$\left(52z^{2}-39z\right)+\left(-4z+3\right)$$

Birinchi guruhda 13z ni va ikkinchi guruhda -1 ni faktordan chiqaring.

$$13z\left(4z-3\right)-\left(4z-3\right)$$

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

$$\left(4z-3\right)\left(13z-1\right)$$

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.

$$52z^{2}-43z+3=0$$

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.

$$z=\frac{-\left(-43\right)±\sqrt{\left(-43\right)^{2}-4\times 52\times 3}}{2\times 52}$$

-43 kvadratini chiqarish.

$$z=\frac{-\left(-43\right)±\sqrt{1849-4\times 52\times 3}}{2\times 52}$$

-4 ni 52 marotabaga ko'paytirish.

$$z=\frac{-\left(-43\right)±\sqrt{1849-208\times 3}}{2\times 52}$$

-208 ni 3 marotabaga ko'paytirish.

$$z=\frac{-\left(-43\right)±\sqrt{1849-624}}{2\times 52}$$

1849 ni -624 ga qo'shish.

$$z=\frac{-\left(-43\right)±\sqrt{1225}}{2\times 52}$$

1225 ning kvadrat ildizini chiqarish.

$$z=\frac{-\left(-43\right)±35}{2\times 52}$$

-43 ning teskarisi 43 ga teng.

$$z=\frac{43±35}{2\times 52}$$

2 ni 52 marotabaga ko'paytirish.

$$z=\frac{43±35}{104}$$

z=\frac{43±35}{104} tenglamasini yeching, bunda ± musbat. 43 ni 35 ga qo'shish.

$$z=\frac{78}{104}$$

\frac{78}{104} ulushini 26 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

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

z=\frac{43±35}{104} tenglamasini yeching, bunda ± manfiy. 43 dan 35 ni ayirish.

$$z=\frac{8}{104}$$

\frac{8}{104} ulushini 8 ni chiqarib, bekor qilish hisobiga eng past shartlarga kamaytiring.

$$z=\frac{1}{13}$$

ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right) formulasi yordamida amalni hisoblang. x_{1} uchun \frac{3}{4} ga va x_{2} uchun \frac{1}{13} ga bo‘ling.

$$52z^{2}-43z+3=52\left(z-\frac{3}{4}\right)\left(z-\frac{1}{13}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{3}{4} ni z dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$52z^{2}-43z+3=52\times \frac{4z-3}{4}\left(z-\frac{1}{13}\right)$$

Umumiy maxrajni topib va suratlarni ayirib \frac{1}{13} ni z dan ayirish. So'ngra imkoni boricha kasrni eng kichik shartga qisqartirish.

$$52z^{2}-43z+3=52\times \frac{4z-3}{4}\times \frac{13z-1}{13}$$

Raqamlash sonlarini va maxraj sonlariga ko'paytirish orqali \frac{4z-3}{4} ni \frac{13z-1}{13} ga ko'paytirish. So'ngra kasrni imkoni boricha eng kam a'zoga qisqartiring.

$$52z^{2}-43z+3=52\times \frac{\left(4z-3\right)\left(13z-1\right)}{4\times 13}$$

4 ni 13 marotabaga ko'paytirish.

$$52z^{2}-43z+3=52\times \frac{\left(4z-3\right)\left(13z-1\right)}{52}$$

52 va 52 ichida eng katta umumiy 52 faktorini bekor qiling.

$$52z^{2}-43z+3=\left(4z-3\right)\left(13z-1\right)$$

Baholash

$$\left(4z-3\right)\left(13z-1\right)$$