Whakaoti i te -16t^2+96t-108

$$- 16 t ^ { 2 } + 96 t - 108$$

Tauwehe

$$4\left(3-2t\right)\left(2t-9\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(-4t^{2}+24t-27\right)$$

Whakaarohia te -4t^{2}+24t-27. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -4t^{2}+at+bt-27. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=24 ab=-4\left(-27\right)=108$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 108.

$$1,108 2,54 3,36 4,27 6,18 9,12$$

Tātaihia te tapeke mō ia takirua.

$$1+108=109 2+54=56 3+36=39 4+27=31 6+18=24 9+12=21$$

Ko te otinga te takirua ka hoatu i te tapeke 24.

$$a=18 b=6$$

Tuhia anō te -4t^{2}+24t-27 hei \left(-4t^{2}+18t\right)+\left(6t-27\right).

$$\left(-4t^{2}+18t\right)+\left(6t-27\right)$$

Tauwehea te -2t i te tuatahi me te 3 i te rōpū tuarua.

$$-2t\left(2t-9\right)+3\left(2t-9\right)$$

Whakatauwehea atu te kīanga pātahi 2t-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2t-9\right)\left(-2t+3\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(2t-9\right)\left(-2t+3\right)$$

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.

$$-16t^{2}+96t-108=0$$

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.

$$t=\frac{-96±\sqrt{96^{2}-4\left(-16\right)\left(-108\right)}}{2\left(-16\right)}$$

Pūrua 96.

$$t=\frac{-96±\sqrt{9216-4\left(-16\right)\left(-108\right)}}{2\left(-16\right)}$$

Whakareatia -4 ki te -16.

$$t=\frac{-96±\sqrt{9216+64\left(-108\right)}}{2\left(-16\right)}$$

Whakareatia 64 ki te -108.

$$t=\frac{-96±\sqrt{9216-6912}}{2\left(-16\right)}$$

Tāpiri 9216 ki te -6912.

$$t=\frac{-96±\sqrt{2304}}{2\left(-16\right)}$$

Tuhia te pūtakerua o te 2304.

$$t=\frac{-96±48}{2\left(-16\right)}$$

Whakareatia 2 ki te -16.

$$t=\frac{-96±48}{-32}$$

Nā, me whakaoti te whārite t=\frac{-96±48}{-32} ina he tāpiri te ±. Tāpiri -96 ki te 48.

$$t=-\frac{48}{-32}$$

Whakahekea te hautanga \frac{-48}{-32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

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

Nā, me whakaoti te whārite t=\frac{-96±48}{-32} ina he tango te ±. Tango 48 mai i -96.

$$t=-\frac{144}{-32}$$

Whakahekea te hautanga \frac{-144}{-32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$t=\frac{9}{2}$$

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}{2} mō te x_{1} me te \frac{9}{2} mō te x_{2}.

$$-16t^{2}+96t-108=-16\left(t-\frac{3}{2}\right)\left(t-\frac{9}{2}\right)$$

Tango \frac{3}{2} mai i t 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.

$$-16t^{2}+96t-108=-16\times \frac{-2t+3}{-2}\left(t-\frac{9}{2}\right)$$

Tango \frac{9}{2} mai i t 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.

$$-16t^{2}+96t-108=-16\times \frac{-2t+3}{-2}\times \frac{-2t+9}{-2}$$

Whakareatia \frac{-2t+3}{-2} ki te \frac{-2t+9}{-2} 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.

$$-16t^{2}+96t-108=-16\times \frac{\left(-2t+3\right)\left(-2t+9\right)}{-2\left(-2\right)}$$

Whakareatia -2 ki te -2.

$$-16t^{2}+96t-108=-16\times \frac{\left(-2t+3\right)\left(-2t+9\right)}{4}$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te -16 me te 4.

$$-16t^{2}+96t-108=-4\left(-2t+3\right)\left(-2t+9\right)$$

Aromātai

$$-16t^{2}+96t-108$$