Whakaoti i te t^2:2^4-6t-2^8=0

$$t ^ { 2 } : 2 ^ { 4 } - 6 t - 2 ^ { 8 } = 0$$

Whakaoti mō t

$$t=-32$$

$$t=128$$

Ngā Upane Whakamahi Tauwehe

Tātaihia te 2 mā te pū o 4, kia riro ko 16.

$$\frac{t^{2}}{16}-6t-2^{8}=0$$

Tātaihia te 2 mā te pū o 8, kia riro ko 256.

$$\frac{t^{2}}{16}-6t-256=0$$

Whakareatia ngā taha e rua o te whārite ki te 16.

$$t^{2}-96t-4096=0$$

Hei whakaoti i te whārite, whakatauwehea te t^{2}-96t-4096 mā te whakamahi i te tātai t^{2}+\left(a+b\right)t+ab=\left(t+a\right)\left(t+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-96 ab=-4096$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -4096.

$$1,-4096 2,-2048 4,-1024 8,-512 16,-256 32,-128 64,-64$$

Tātaihia te tapeke mō ia takirua.

$$1-4096=-4095 2-2048=-2046 4-1024=-1020 8-512=-504 16-256=-240 32-128=-96 64-64=0$$

Ko te otinga te takirua ka hoatu i te tapeke -96.

$$a=-128 b=32$$

Me tuhi anō te kīanga whakatauwehe \left(t+a\right)\left(t+b\right) mā ngā uara i tātaihia.

$$\left(t-128\right)\left(t+32\right)$$

Hei kimi otinga whārite, me whakaoti te t-128=0 me te t+32=0.

$$t=128 t=-32$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tātaihia te 2 mā te pū o 4, kia riro ko 16.

$$\frac{t^{2}}{16}-6t-2^{8}=0$$

Tātaihia te 2 mā te pū o 8, kia riro ko 256.

$$\frac{t^{2}}{16}-6t-256=0$$

Whakareatia ngā taha e rua o te whārite ki te 16.

$$t^{2}-96t-4096=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei t^{2}+at+bt-4096. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-96 ab=1\left(-4096\right)=-4096$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -4096.

$$1,-4096 2,-2048 4,-1024 8,-512 16,-256 32,-128 64,-64$$

Tātaihia te tapeke mō ia takirua.

$$1-4096=-4095 2-2048=-2046 4-1024=-1020 8-512=-504 16-256=-240 32-128=-96 64-64=0$$

Ko te otinga te takirua ka hoatu i te tapeke -96.

$$a=-128 b=32$$

Tuhia anō te t^{2}-96t-4096 hei \left(t^{2}-128t\right)+\left(32t-4096\right).

$$\left(t^{2}-128t\right)+\left(32t-4096\right)$$

Tauwehea te t i te tuatahi me te 32 i te rōpū tuarua.

$$t\left(t-128\right)+32\left(t-128\right)$$

Whakatauwehea atu te kīanga pātahi t-128 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(t-128\right)\left(t+32\right)$$

Hei kimi otinga whārite, me whakaoti te t-128=0 me te t+32=0.

$$t=128 t=-32$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 2 mā te pū o 4, kia riro ko 16.

$$\frac{t^{2}}{16}-6t-2^{8}=0$$

Tātaihia te 2 mā te pū o 8, kia riro ko 256.

$$\frac{t^{2}}{16}-6t-256=0$$

Whakareatia ngā taha e rua o te whārite ki te 16.

$$t^{2}-96t-4096=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -96 mō b, me -4096 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-96\right)±\sqrt{\left(-96\right)^{2}-4\left(-4096\right)}}{2}$$

Pūrua -96.

$$t=\frac{-\left(-96\right)±\sqrt{9216-4\left(-4096\right)}}{2}$$

Whakareatia -4 ki te -4096.

$$t=\frac{-\left(-96\right)±\sqrt{9216+16384}}{2}$$

Tāpiri 9216 ki te 16384.

$$t=\frac{-\left(-96\right)±\sqrt{25600}}{2}$$

Tuhia te pūtakerua o te 25600.

$$t=\frac{-\left(-96\right)±160}{2}$$

Ko te tauaro o -96 ko 96.

$$t=\frac{96±160}{2}$$

Nā, me whakaoti te whārite t=\frac{96±160}{2} ina he tāpiri te ±. Tāpiri 96 ki te 160.

$$t=\frac{256}{2}$$

Whakawehe 256 ki te 2.

$$t=128$$

Nā, me whakaoti te whārite t=\frac{96±160}{2} ina he tango te ±. Tango 160 mai i 96.

$$t=-\frac{64}{2}$$

Whakawehe -64 ki te 2.

$$t=-32$$

Kua oti te whārite te whakatau.

$$t=128 t=-32$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 2 mā te pū o 4, kia riro ko 16.

$$\frac{t^{2}}{16}-6t-2^{8}=0$$

Tātaihia te 2 mā te pū o 8, kia riro ko 256.

$$\frac{t^{2}}{16}-6t-256=0$$

Me tāpiri te 256 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$\frac{t^{2}}{16}-6t=256$$

Whakareatia ngā taha e rua o te whārite ki te 16.

$$t^{2}-96t=4096$$

Whakawehea te -96, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -48. Nā, tāpiria te pūrua o te -48 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-96t+\left(-48\right)^{2}=4096+\left(-48\right)^{2}$$

Pūrua -48.

$$t^{2}-96t+2304=4096+2304$$

Tāpiri 4096 ki te 2304.

$$t^{2}-96t+2304=6400$$

Tauwehea t^{2}-96t+2304. 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}.

$$\left(t-48\right)^{2}=6400$$

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

$$\sqrt{\left(t-48\right)^{2}}=\sqrt{6400}$$

Whakarūnātia.

$$t-48=80 t-48=-80$$

Me tāpiri 48 ki ngā taha e rua o te whārite.

$$t=128 t=-32$$