Whakaoti i te 16z^2-54z+35

$$16 z ^ { 2 } - 54 z + 35$$

Tauwehe

$$\left(2z-5\right)\left(8z-7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 16z^{2}+az+bz+35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-54 ab=16\times 35=560$$

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 560.

$$-1,-560 -2,-280 -4,-140 -5,-112 -7,-80 -8,-70 -10,-56 -14,-40 -16,-35 -20,-28$$

Tātaihia te tapeke mō ia takirua.

$$-1-560=-561 -2-280=-282 -4-140=-144 -5-112=-117 -7-80=-87 -8-70=-78 -10-56=-66 -14-40=-54 -16-35=-51 -20-28=-48$$

Ko te otinga te takirua ka hoatu i te tapeke -54.

$$a=-40 b=-14$$

Tuhia anō te 16z^{2}-54z+35 hei \left(16z^{2}-40z\right)+\left(-14z+35\right).

$$\left(16z^{2}-40z\right)+\left(-14z+35\right)$$

Tauwehea te 8z i te tuatahi me te -7 i te rōpū tuarua.

$$8z\left(2z-5\right)-7\left(2z-5\right)$$

Whakatauwehea atu te kīanga pātahi 2z-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2z-5\right)\left(8z-7\right)$$

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.

$$16z^{2}-54z+35=0$$

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.

$$z=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 16\times 35}}{2\times 16}$$

Pūrua -54.

$$z=\frac{-\left(-54\right)±\sqrt{2916-4\times 16\times 35}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$z=\frac{-\left(-54\right)±\sqrt{2916-64\times 35}}{2\times 16}$$

Whakareatia -64 ki te 35.

$$z=\frac{-\left(-54\right)±\sqrt{2916-2240}}{2\times 16}$$

Tāpiri 2916 ki te -2240.

$$z=\frac{-\left(-54\right)±\sqrt{676}}{2\times 16}$$

Tuhia te pūtakerua o te 676.

$$z=\frac{-\left(-54\right)±26}{2\times 16}$$

Ko te tauaro o -54 ko 54.

$$z=\frac{54±26}{2\times 16}$$

Whakareatia 2 ki te 16.

$$z=\frac{54±26}{32}$$

Nā, me whakaoti te whārite z=\frac{54±26}{32} ina he tāpiri te ±. Tāpiri 54 ki te 26.

$$z=\frac{80}{32}$$

Whakahekea te hautanga \frac{80}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$z=\frac{5}{2}$$

Nā, me whakaoti te whārite z=\frac{54±26}{32} ina he tango te ±. Tango 26 mai i 54.

$$z=\frac{28}{32}$$

Whakahekea te hautanga \frac{28}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$z=\frac{7}{8}$$

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{5}{2} mō te x_{1} me te \frac{7}{8} mō te x_{2}.

$$16z^{2}-54z+35=16\left(z-\frac{5}{2}\right)\left(z-\frac{7}{8}\right)$$

Tango \frac{5}{2} 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.

$$16z^{2}-54z+35=16\times \frac{2z-5}{2}\left(z-\frac{7}{8}\right)$$

Tango \frac{7}{8} 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.

$$16z^{2}-54z+35=16\times \frac{2z-5}{2}\times \frac{8z-7}{8}$$

Whakareatia \frac{2z-5}{2} ki te \frac{8z-7}{8} 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.

$$16z^{2}-54z+35=16\times \frac{\left(2z-5\right)\left(8z-7\right)}{2\times 8}$$

Whakareatia 2 ki te 8.

$$16z^{2}-54z+35=16\times \frac{\left(2z-5\right)\left(8z-7\right)}{16}$$

Whakakorea atu te tauwehe pūnoa nui rawa 16 i roto i te 16 me te 16.

$$16z^{2}-54z+35=\left(2z-5\right)\left(8z-7\right)$$

Aromātai

$$\left(2z-5\right)\left(8z-7\right)$$