Whakaoti i te t^2-24t-180=0

$$t ^ { 2 } - 24 t - 180 = 0$$

Whakaoti mō t

$$t=-6$$

$$t=30$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te t^{2}-24t-180 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.

$$a+b=-24 ab=-180$$

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 -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -24.

$$a=-30 b=6$$

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

$$\left(t-30\right)\left(t+6\right)$$

Hei kimi otinga whārite, me whakaoti te t-30=0 me te t+6=0.

$$t=30 t=-6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei t^{2}+at+bt-180. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-24 ab=1\left(-180\right)=-180$$

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 -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -24.

$$a=-30 b=6$$

Tuhia anō te t^{2}-24t-180 hei \left(t^{2}-30t\right)+\left(6t-180\right).

$$\left(t^{2}-30t\right)+\left(6t-180\right)$$

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

$$t\left(t-30\right)+6\left(t-30\right)$$

Whakatauwehea atu te kīanga pātahi t-30 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(t-30\right)\left(t+6\right)$$

Hei kimi otinga whārite, me whakaoti te t-30=0 me te t+6=0.

$$t=30 t=-6$$

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.

$$t^{2}-24t-180=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -24 mō b, me -180 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\left(-180\right)}}{2}$$

Pūrua -24.

$$t=\frac{-\left(-24\right)±\sqrt{576-4\left(-180\right)}}{2}$$

Whakareatia -4 ki te -180.

$$t=\frac{-\left(-24\right)±\sqrt{576+720}}{2}$$

Tāpiri 576 ki te 720.

$$t=\frac{-\left(-24\right)±\sqrt{1296}}{2}$$

Tuhia te pūtakerua o te 1296.

$$t=\frac{-\left(-24\right)±36}{2}$$

Ko te tauaro o -24 ko 24.

$$t=\frac{24±36}{2}$$

Nā, me whakaoti te whārite t=\frac{24±36}{2} ina he tāpiri te ±. Tāpiri 24 ki te 36.

$$t=\frac{60}{2}$$

Whakawehe 60 ki te 2.

$$t=30$$

Nā, me whakaoti te whārite t=\frac{24±36}{2} ina he tango te ±. Tango 36 mai i 24.

$$t=-\frac{12}{2}$$

Whakawehe -12 ki te 2.

$$t=-6$$

Kua oti te whārite te whakatau.

$$t=30 t=-6$$

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.

$$t^{2}-24t-180=0$$

Me tāpiri 180 ki ngā taha e rua o te whārite.

$$t^{2}-24t-180-\left(-180\right)=-\left(-180\right)$$

Mā te tango i te -180 i a ia ake anō ka toe ko te 0.

$$t^{2}-24t=-\left(-180\right)$$

Tango -180 mai i 0.

$$t^{2}-24t=180$$

Whakawehea te -24, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -12. Nā, tāpiria te pūrua o te -12 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-24t+\left(-12\right)^{2}=180+\left(-12\right)^{2}$$

Pūrua -12.

$$t^{2}-24t+144=180+144$$

Tāpiri 180 ki te 144.

$$t^{2}-24t+144=324$$

Tauwehea t^{2}-24t+144. 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}.

$$\left(t-12\right)^{2}=324$$

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

$$\sqrt{\left(t-12\right)^{2}}=\sqrt{324}$$

Whakarūnātia.

$$t-12=18 t-12=-18$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$t=30 t=-6$$