Whakaoti i te 5z^2-33z+18

$$5 z ^ { 2 } - 33 z + 18$$

Tauwehe

$$\left(z-6\right)\left(5z-3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 5z^{2}+az+bz+18. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-33 ab=5\times 18=90$$

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 90.

$$-1,-90 -2,-45 -3,-30 -5,-18 -6,-15 -9,-10$$

Tātaihia te tapeke mō ia takirua.

$$-1-90=-91 -2-45=-47 -3-30=-33 -5-18=-23 -6-15=-21 -9-10=-19$$

Ko te otinga te takirua ka hoatu i te tapeke -33.

$$a=-30 b=-3$$

Tuhia anō te 5z^{2}-33z+18 hei \left(5z^{2}-30z\right)+\left(-3z+18\right).

$$\left(5z^{2}-30z\right)+\left(-3z+18\right)$$

Tauwehea te 5z i te tuatahi me te -3 i te rōpū tuarua.

$$5z\left(z-6\right)-3\left(z-6\right)$$

Whakatauwehea atu te kīanga pātahi z-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(z-6\right)\left(5z-3\right)$$

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.

$$5z^{2}-33z+18=0$$

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.

$$z=\frac{-\left(-33\right)±\sqrt{\left(-33\right)^{2}-4\times 5\times 18}}{2\times 5}$$

Pūrua -33.

$$z=\frac{-\left(-33\right)±\sqrt{1089-4\times 5\times 18}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$z=\frac{-\left(-33\right)±\sqrt{1089-20\times 18}}{2\times 5}$$

Whakareatia -20 ki te 18.

$$z=\frac{-\left(-33\right)±\sqrt{1089-360}}{2\times 5}$$

Tāpiri 1089 ki te -360.

$$z=\frac{-\left(-33\right)±\sqrt{729}}{2\times 5}$$

Tuhia te pūtakerua o te 729.

$$z=\frac{-\left(-33\right)±27}{2\times 5}$$

Ko te tauaro o -33 ko 33.

$$z=\frac{33±27}{2\times 5}$$

Whakareatia 2 ki te 5.

$$z=\frac{33±27}{10}$$

Nā, me whakaoti te whārite z=\frac{33±27}{10} ina he tāpiri te ±. Tāpiri 33 ki te 27.

$$z=\frac{60}{10}$$

Whakawehe 60 ki te 10.

$$z=6$$

Nā, me whakaoti te whārite z=\frac{33±27}{10} ina he tango te ±. Tango 27 mai i 33.

$$z=\frac{6}{10}$$

Whakahekea te hautanga \frac{6}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$z=\frac{3}{5}$$

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 6 mō te x_{1} me te \frac{3}{5} mō te x_{2}.

$$5z^{2}-33z+18=5\left(z-6\right)\left(z-\frac{3}{5}\right)$$

Tango \frac{3}{5} 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.

$$5z^{2}-33z+18=5\left(z-6\right)\times \frac{5z-3}{5}$$

Whakakorea atu te tauwehe pūnoa nui rawa 5 i roto i te 5 me te 5.

$$5z^{2}-33z+18=\left(z-6\right)\left(5z-3\right)$$

Aromātai

$$\left(z-6\right)\left(5z-3\right)$$