Whakaoti i te 4t^2+3t=1

$$4 { t }^{ 2 } +3t = 1$$

Whakaoti mō t

$$t=-1$$

$$t=\frac{1}{4}=0.25$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 1 mai i ngā taha e rua.

$$4t^{2}+3t-1=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4t^{2}+at+bt-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=3 ab=4\left(-1\right)=-4$$

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 -4.

$$-1,4 -2,2$$

Tātaihia te tapeke mō ia takirua.

$$-1+4=3 -2+2=0$$

Ko te otinga te takirua ka hoatu i te tapeke 3.

$$a=-1 b=4$$

Tuhia anō te 4t^{2}+3t-1 hei \left(4t^{2}-t\right)+\left(4t-1\right).

$$\left(4t^{2}-t\right)+\left(4t-1\right)$$

Whakatauwehea atu t i te 4t^{2}-t.

$$t\left(4t-1\right)+4t-1$$

Whakatauwehea atu te kīanga pātahi 4t-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4t-1\right)\left(t+1\right)$$

Hei kimi otinga whārite, me whakaoti te 4t-1=0 me te t+1=0.

$$t=\frac{1}{4} t=-1$$

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.

$$4t^{2}+3t=1$$

Me tango 1 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 1 i a ia ake anō ka toe ko te 0.

$$4t^{2}+3t-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 3 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua 3.

$$t=\frac{-3±\sqrt{9-4\times 4\left(-1\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$t=\frac{-3±\sqrt{9-16\left(-1\right)}}{2\times 4}$$

Whakareatia -16 ki te -1.

$$t=\frac{-3±\sqrt{9+16}}{2\times 4}$$

Tāpiri 9 ki te 16.

$$t=\frac{-3±\sqrt{25}}{2\times 4}$$

Tuhia te pūtakerua o te 25.

$$t=\frac{-3±5}{2\times 4}$$

Whakareatia 2 ki te 4.

$$t=\frac{-3±5}{8}$$

Nā, me whakaoti te whārite t=\frac{-3±5}{8} ina he tāpiri te ±. Tāpiri -3 ki te 5.

$$t=\frac{2}{8}$$

Whakahekea te hautanga \frac{2}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$t=\frac{1}{4}$$

Nā, me whakaoti te whārite t=\frac{-3±5}{8} ina he tango te ±. Tango 5 mai i -3.

$$t=-\frac{8}{8}$$

Whakawehe -8 ki te 8.

$$t=-1$$

Kua oti te whārite te whakatau.

$$t=\frac{1}{4} t=-1$$

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.

$$4t^{2}+3t=1$$

Whakawehea ngā taha e rua ki te 4.

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

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

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

Whakawehea te \frac{3}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{8}. Nā, tāpiria te pūrua o te \frac{3}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}+\frac{3}{4}t+\left(\frac{3}{8}\right)^{2}=\frac{1}{4}+\left(\frac{3}{8}\right)^{2}$$

Pūruatia \frac{3}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}+\frac{3}{4}t+\frac{9}{64}=\frac{1}{4}+\frac{9}{64}$$

Tāpiri \frac{1}{4} ki te \frac{9}{64} 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.

$$t^{2}+\frac{3}{4}t+\frac{9}{64}=\frac{25}{64}$$

Tauwehea te t^{2}+\frac{3}{4}t+\frac{9}{64}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(t+\frac{3}{8}\right)^{2}=\frac{25}{64}$$

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

$$\sqrt{\left(t+\frac{3}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Whakarūnātia.

$$t+\frac{3}{8}=\frac{5}{8} t+\frac{3}{8}=-\frac{5}{8}$$

Me tango \frac{3}{8} mai i ngā taha e rua o te whārite.

$$t=\frac{1}{4} t=-1$$