Whakaoti i te (1/2)32t^2+16t=480

$$( 1 / 2 ) 32 t ^ { 2 } + 16 t = 480$$

Whakaoti mō t

$$t=-6$$

$$t=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakareatia te \frac{1}{2} ki te 32, ka 16.

$$16t^{2}+16t=480$$

Tangohia te 480 mai i ngā taha e rua.

$$16t^{2}+16t-480=0$$

Whakawehea ngā taha e rua ki te 16.

$$t^{2}+t-30=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei t^{2}+at+bt-30. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=1\left(-30\right)=-30$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=-5 b=6$$

Tuhia anō te t^{2}+t-30 hei \left(t^{2}-5t\right)+\left(6t-30\right).

$$\left(t^{2}-5t\right)+\left(6t-30\right)$$

Tauwehea te t i te tuatahi me te 6 i te rōpū tuarua.

$$t\left(t-5\right)+6\left(t-5\right)$$

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

$$\left(t-5\right)\left(t+6\right)$$

Hei kimi otinga whārite, me whakaoti te t-5=0 me te t+6=0.

$$t=5 t=-6$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakareatia te \frac{1}{2} ki te 32, ka 16.

$$16t^{2}+16t=480$$

Tangohia te 480 mai i ngā taha e rua.

$$16t^{2}+16t-480=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, 16 mō b, me -480 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-16±\sqrt{16^{2}-4\times 16\left(-480\right)}}{2\times 16}$$

Pūrua 16.

$$t=\frac{-16±\sqrt{256-4\times 16\left(-480\right)}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$t=\frac{-16±\sqrt{256-64\left(-480\right)}}{2\times 16}$$

Whakareatia -64 ki te -480.

$$t=\frac{-16±\sqrt{256+30720}}{2\times 16}$$

Tāpiri 256 ki te 30720.

$$t=\frac{-16±\sqrt{30976}}{2\times 16}$$

Tuhia te pūtakerua o te 30976.

$$t=\frac{-16±176}{2\times 16}$$

Whakareatia 2 ki te 16.

$$t=\frac{-16±176}{32}$$

Nā, me whakaoti te whārite t=\frac{-16±176}{32} ina he tāpiri te ±. Tāpiri -16 ki te 176.

$$t=\frac{160}{32}$$

Whakawehe 160 ki te 32.

$$t=5$$

Nā, me whakaoti te whārite t=\frac{-16±176}{32} ina he tango te ±. Tango 176 mai i -16.

$$t=-\frac{192}{32}$$

Whakawehe -192 ki te 32.

$$t=-6$$

Kua oti te whārite te whakatau.

$$t=5 t=-6$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakareatia te \frac{1}{2} ki te 32, ka 16.

$$16t^{2}+16t=480$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16t^{2}+16t}{16}=\frac{480}{16}$$

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

$$t^{2}+\frac{16}{16}t=\frac{480}{16}$$

Whakawehe 16 ki te 16.

$$t^{2}+t=\frac{480}{16}$$

Whakawehe 480 ki te 16.

$$t^{2}+t=30$$

Whakawehea te 1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{2}. Nā, tāpiria te pūrua o te \frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}+t+\left(\frac{1}{2}\right)^{2}=30+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}+t+\frac{1}{4}=30+\frac{1}{4}$$

Tāpiri 30 ki te \frac{1}{4}.

$$t^{2}+t+\frac{1}{4}=\frac{121}{4}$$

Tauwehea t^{2}+t+\frac{1}{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}.

$$\left(t+\frac{1}{2}\right)^{2}=\frac{121}{4}$$

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

$$\sqrt{\left(t+\frac{1}{2}\right)^{2}}=\sqrt{\frac{121}{4}}$$

Whakarūnātia.

$$t+\frac{1}{2}=\frac{11}{2} t+\frac{1}{2}=-\frac{11}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$t=5 t=-6$$