Whakaoti i te t^2+3t/2=t+7/4

$$\frac { t ^ { 2 } + 3 t } { 2 } = \frac { t + 7 } { 4 }$$

Whakaoti mō t

$$t = -\frac{7}{2} = -3\frac{1}{2} = -3.5$$

$$t=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me whakarea ngā taha e rua o te whārite ki te 4, arā, te tauraro pātahi he tino iti rawa te kitea o 2,4.

$$2\left(t^{2}+3t\right)=t+7$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te t^{2}+3t.

$$2t^{2}+6t=t+7$$

Tangohia te t mai i ngā taha e rua.

$$2t^{2}+6t-t=7$$

Pahekotia te 6t me -t, ka 5t.

$$2t^{2}+5t=7$$

Tangohia te 7 mai i ngā taha e rua.

$$2t^{2}+5t-7=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2t^{2}+at+bt-7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=2\left(-7\right)=-14$$

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 -14.

$$-1,14 -2,7$$

Tātaihia te tapeke mō ia takirua.

$$-1+14=13 -2+7=5$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=-2 b=7$$

Tuhia anō te 2t^{2}+5t-7 hei \left(2t^{2}-2t\right)+\left(7t-7\right).

$$\left(2t^{2}-2t\right)+\left(7t-7\right)$$

Tauwehea te 2t i te tuatahi me te 7 i te rōpū tuarua.

$$2t\left(t-1\right)+7\left(t-1\right)$$

Whakatauwehea atu te kīanga pātahi t-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(t-1\right)\left(2t+7\right)$$

Hei kimi otinga whārite, me whakaoti te t-1=0 me te 2t+7=0.

$$t=1 t=-\frac{7}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 4, arā, te tauraro pātahi he tino iti rawa te kitea o 2,4.

$$2\left(t^{2}+3t\right)=t+7$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te t^{2}+3t.

$$2t^{2}+6t=t+7$$

Tangohia te t mai i ngā taha e rua.

$$2t^{2}+6t-t=7$$

Pahekotia te 6t me -t, ka 5t.

$$2t^{2}+5t=7$$

Tangohia te 7 mai i ngā taha e rua.

$$2t^{2}+5t-7=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 5 mō b, me -7 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-5±\sqrt{5^{2}-4\times 2\left(-7\right)}}{2\times 2}$$

Pūrua 5.

$$t=\frac{-5±\sqrt{25-4\times 2\left(-7\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$t=\frac{-5±\sqrt{25-8\left(-7\right)}}{2\times 2}$$

Whakareatia -8 ki te -7.

$$t=\frac{-5±\sqrt{25+56}}{2\times 2}$$

Tāpiri 25 ki te 56.

$$t=\frac{-5±\sqrt{81}}{2\times 2}$$

Tuhia te pūtakerua o te 81.

$$t=\frac{-5±9}{2\times 2}$$

Whakareatia 2 ki te 2.

$$t=\frac{-5±9}{4}$$

Nā, me whakaoti te whārite t=\frac{-5±9}{4} ina he tāpiri te ±. Tāpiri -5 ki te 9.

$$t=\frac{4}{4}$$

Whakawehe 4 ki te 4.

$$t=1$$

Nā, me whakaoti te whārite t=\frac{-5±9}{4} ina he tango te ±. Tango 9 mai i -5.

$$t=-\frac{14}{4}$$

Whakahekea te hautanga \frac{-14}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$t=-\frac{7}{2}$$

Kua oti te whārite te whakatau.

$$t=1 t=-\frac{7}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 4, arā, te tauraro pātahi he tino iti rawa te kitea o 2,4.

$$2\left(t^{2}+3t\right)=t+7$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te t^{2}+3t.

$$2t^{2}+6t=t+7$$

Tangohia te t mai i ngā taha e rua.

$$2t^{2}+6t-t=7$$

Pahekotia te 6t me -t, ka 5t.

$$2t^{2}+5t=7$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2t^{2}+5t}{2}=\frac{7}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$t^{2}+\frac{5}{2}t=\frac{7}{2}$$

Whakawehea te \frac{5}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{4}. Nā, tāpiria te pūrua o te \frac{5}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}+\frac{5}{2}t+\left(\frac{5}{4}\right)^{2}=\frac{7}{2}+\left(\frac{5}{4}\right)^{2}$$

Pūruatia \frac{5}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}+\frac{5}{2}t+\frac{25}{16}=\frac{7}{2}+\frac{25}{16}$$

Tāpiri \frac{7}{2} ki te \frac{25}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$t^{2}+\frac{5}{2}t+\frac{25}{16}=\frac{81}{16}$$

Tauwehea t^{2}+\frac{5}{2}t+\frac{25}{16}. 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}.

$$\left(t+\frac{5}{4}\right)^{2}=\frac{81}{16}$$

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

$$\sqrt{\left(t+\frac{5}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Whakarūnātia.

$$t+\frac{5}{4}=\frac{9}{4} t+\frac{5}{4}=-\frac{9}{4}$$

Me tango \frac{5}{4} mai i ngā taha e rua o te whārite.

$$t=1 t=-\frac{7}{2}$$