Whakaoti i te -16t^2+64t+80=128

$$- 16 t ^ { 2 } + 64 t + 80 = 128$$

Whakaoti mō t

$$t=1$$

$$t=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 128 mai i ngā taha e rua.

$$-16t^{2}+64t+80-128=0$$

Tangohia te 128 i te 80, ka -48.

$$-16t^{2}+64t-48=0$$

Whakawehea ngā taha e rua ki te 16.

$$-t^{2}+4t-3=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -t^{2}+at+bt-3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=-\left(-3\right)=3$$

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.

$$a=3 b=1$$

Tuhia anō te -t^{2}+4t-3 hei \left(-t^{2}+3t\right)+\left(t-3\right).

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

Whakatauwehea atu -t i te -t^{2}+3t.

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

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

$$\left(t-3\right)\left(-t+1\right)$$

Hei kimi otinga whārite, me whakaoti te t-3=0 me te -t+1=0.

$$t=3 t=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$-16t^{2}+64t+80=128$$

Me tango 128 mai i ngā taha e rua o te whārite.

$$-16t^{2}+64t+80-128=128-128$$

Mā te tango i te 128 i a ia ake anō ka toe ko te 0.

$$-16t^{2}+64t+80-128=0$$

Tango 128 mai i 80.

$$-16t^{2}+64t-48=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -16 mō a, 64 mō b, me -48 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-64±\sqrt{64^{2}-4\left(-16\right)\left(-48\right)}}{2\left(-16\right)}$$

Pūrua 64.

$$t=\frac{-64±\sqrt{4096-4\left(-16\right)\left(-48\right)}}{2\left(-16\right)}$$

Whakareatia -4 ki te -16.

$$t=\frac{-64±\sqrt{4096+64\left(-48\right)}}{2\left(-16\right)}$$

Whakareatia 64 ki te -48.

$$t=\frac{-64±\sqrt{4096-3072}}{2\left(-16\right)}$$

Tāpiri 4096 ki te -3072.

$$t=\frac{-64±\sqrt{1024}}{2\left(-16\right)}$$

Tuhia te pūtakerua o te 1024.

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

Whakareatia 2 ki te -16.

$$t=\frac{-64±32}{-32}$$

Nā, me whakaoti te whārite t=\frac{-64±32}{-32} ina he tāpiri te ±. Tāpiri -64 ki te 32.

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

Whakawehe -32 ki te -32.

$$t=1$$

Nā, me whakaoti te whārite t=\frac{-64±32}{-32} ina he tango te ±. Tango 32 mai i -64.

$$t=-\frac{96}{-32}$$

Whakawehe -96 ki te -32.

$$t=3$$

Kua oti te whārite te whakatau.

$$t=1 t=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-16t^{2}+64t+80=128$$

Me tango 80 mai i ngā taha e rua o te whārite.

$$-16t^{2}+64t+80-80=128-80$$

Mā te tango i te 80 i a ia ake anō ka toe ko te 0.

$$-16t^{2}+64t=128-80$$

Tango 80 mai i 128.

$$-16t^{2}+64t=48$$

Whakawehea ngā taha e rua ki te -16.

$$\frac{-16t^{2}+64t}{-16}=\frac{48}{-16}$$

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

$$t^{2}+\frac{64}{-16}t=\frac{48}{-16}$$

Whakawehe 64 ki te -16.

$$t^{2}-4t=\frac{48}{-16}$$

Whakawehe 48 ki te -16.

$$t^{2}-4t=-3$$

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūrua -2.

$$t^{2}-4t+4=-3+4$$

Tāpiri -3 ki te 4.

$$t^{2}-4t+4=1$$

Tauwehea t^{2}-4t+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}.

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

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

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

Whakarūnātia.

$$t-2=1 t-2=-1$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$t=3 t=1$$