Whakaoti i te 0=-16t^2+48t-32

$$0 = - 16 t ^ { 2 } + 48 t - 32$$

Whakaoti mō t

$$t=1$$

$$t=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-16t^{2}+48t-32=0$$

Whakawehea ngā taha e rua ki te 16.

$$-t^{2}+3t-2=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -t^{2}+at+bt-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=-\left(-2\right)=2$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=2 b=1$$

Tuhia anō te -t^{2}+3t-2 hei \left(-t^{2}+2t\right)+\left(t-2\right).

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

Whakatauwehea atu -t i te -t^{2}+2t.

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

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

$$\left(t-2\right)\left(-t+1\right)$$

Hei kimi otinga whārite, me whakaoti te t-2=0 me te -t+1=0.

$$t=2 t=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-16t^{2}+48t-32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -16 mō a, 48 mō b, me -32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-48±\sqrt{48^{2}-4\left(-16\right)\left(-32\right)}}{2\left(-16\right)}$$

Pūrua 48.

$$t=\frac{-48±\sqrt{2304-4\left(-16\right)\left(-32\right)}}{2\left(-16\right)}$$

Whakareatia -4 ki te -16.

$$t=\frac{-48±\sqrt{2304+64\left(-32\right)}}{2\left(-16\right)}$$

Whakareatia 64 ki te -32.

$$t=\frac{-48±\sqrt{2304-2048}}{2\left(-16\right)}$$

Tāpiri 2304 ki te -2048.

$$t=\frac{-48±\sqrt{256}}{2\left(-16\right)}$$

Tuhia te pūtakerua o te 256.

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

Whakareatia 2 ki te -16.

$$t=\frac{-48±16}{-32}$$

Nā, me whakaoti te whārite t=\frac{-48±16}{-32} ina he tāpiri te ±. Tāpiri -48 ki te 16.

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

Whakawehe -32 ki te -32.

$$t=1$$

Nā, me whakaoti te whārite t=\frac{-48±16}{-32} ina he tango te ±. Tango 16 mai i -48.

$$t=-\frac{64}{-32}$$

Whakawehe -64 ki te -32.

$$t=2$$

Kua oti te whārite te whakatau.

$$t=1 t=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-16t^{2}+48t-32=0$$

Me tāpiri te 32 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$-16t^{2}+48t=32$$

Whakawehea ngā taha e rua ki te -16.

$$\frac{-16t^{2}+48t}{-16}=\frac{32}{-16}$$

Mā te whakawehe ki te -16 ka wetekia te whakareanga ki te -16.

$$t^{2}+\frac{48}{-16}t=\frac{32}{-16}$$

Whakawehe 48 ki te -16.

$$t^{2}-3t=\frac{32}{-16}$$

Whakawehe 32 ki te -16.

$$t^{2}-3t=-2$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-3t+\frac{9}{4}=-2+\frac{9}{4}$$

Tāpiri -2 ki te \frac{9}{4}.

$$t^{2}-3t+\frac{9}{4}=\frac{1}{4}$$

Tauwehea t^{2}-3t+\frac{9}{4}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(t-\frac{3}{2}\right)^{2}=\frac{1}{4}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(t-\frac{3}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$t-\frac{3}{2}=\frac{1}{2} t-\frac{3}{2}=-\frac{1}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$t=2 t=1$$