Whakaoti i te 52z^2-43z+3

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

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 52z^{2}+az+bz+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

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

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 156.

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

Tātaihia te tapeke mō ia takirua.

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

Ko te otinga te takirua ka hoatu i te tapeke -43.

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

Tuhia anō te 52z^{2}-43z+3 hei \left(52z^{2}-39z\right)+\left(-4z+3\right).

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

Tauwehea te 13z i te tuatahi me te -1 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi 4z-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

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

Ko ngā whārite katoa o te āhua ax^{2}+bx+c=0 ka taea te whakaoti mā te whakamahi i te tikanga tātai pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. E rua ngā otinga ka puta i te tikanga tātai pūrua, ko tētahi ina he tāpiri a ±, ā, ko tētahi ina he tango.

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

Pūrua -43.

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

Whakareatia -4 ki te 52.

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

Whakareatia -208 ki te 3.

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

Tāpiri 1849 ki te -624.

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

Tuhia te pūtakerua o te 1225.

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

Ko te tauaro o -43 ko 43.

$$z=\frac{43±35}{2\times 52}$$

Whakareatia 2 ki te 52.

$$z=\frac{43±35}{104}$$

Nā, me whakaoti te whārite z=\frac{43±35}{104} ina he tāpiri te ±. Tāpiri 43 ki te 35.

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

Whakahekea te hautanga \frac{78}{104} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 26.

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

Nā, me whakaoti te whārite z=\frac{43±35}{104} ina he tango te ±. Tango 35 mai i 43.

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

Whakahekea te hautanga \frac{8}{104} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te \frac{3}{4} mō te x_{1} me te \frac{1}{13} mō te x_{2}.

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

Tango \frac{3}{4} mai i z mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Tango \frac{1}{13} mai i z mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Whakareatia \frac{4z-3}{4} ki te \frac{13z-1}{13} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

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

Whakareatia 4 ki te 13.

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

Whakakorea atu te tauwehe pūnoa nui rawa 52 i roto i te 52 me te 52.

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

Aromātai

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