Whakaoti i te 0=-5250+75t^2-225t

$$0 = - 5250 + 75 t ^ { 2 } - 225 t$$

Whakaoti mō t

$$t=-7$$

$$t=10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-5250+75t^{2}-225t=0$$

Whakawehea ngā taha e rua ki te 75.

$$-70+t^{2}-3t=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$t^{2}-3t-70=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei t^{2}+at+bt-70. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-70\right)=-70$$

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 -70.

$$1,-70 2,-35 5,-14 7,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-70=-69 2-35=-33 5-14=-9 7-10=-3$$

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

$$a=-10 b=7$$

Tuhia anō te t^{2}-3t-70 hei \left(t^{2}-10t\right)+\left(7t-70\right).

$$\left(t^{2}-10t\right)+\left(7t-70\right)$$

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

$$t\left(t-10\right)+7\left(t-10\right)$$

Whakatauwehea atu te kīanga pātahi t-10 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(t-10\right)\left(t+7\right)$$

Hei kimi otinga whārite, me whakaoti te t-10=0 me te t+7=0.

$$t=10 t=-7$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-5250+75t^{2}-225t=0$$

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.

$$75t^{2}-225t-5250=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 75 mō a, -225 mō b, me -5250 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-\left(-225\right)±\sqrt{\left(-225\right)^{2}-4\times 75\left(-5250\right)}}{2\times 75}$$

Pūrua -225.

$$t=\frac{-\left(-225\right)±\sqrt{50625-4\times 75\left(-5250\right)}}{2\times 75}$$

Whakareatia -4 ki te 75.

$$t=\frac{-\left(-225\right)±\sqrt{50625-300\left(-5250\right)}}{2\times 75}$$

Whakareatia -300 ki te -5250.

$$t=\frac{-\left(-225\right)±\sqrt{50625+1575000}}{2\times 75}$$

Tāpiri 50625 ki te 1575000.

$$t=\frac{-\left(-225\right)±\sqrt{1625625}}{2\times 75}$$

Tuhia te pūtakerua o te 1625625.

$$t=\frac{-\left(-225\right)±1275}{2\times 75}$$

Ko te tauaro o -225 ko 225.

$$t=\frac{225±1275}{2\times 75}$$

Whakareatia 2 ki te 75.

$$t=\frac{225±1275}{150}$$

Nā, me whakaoti te whārite t=\frac{225±1275}{150} ina he tāpiri te ±. Tāpiri 225 ki te 1275.

$$t=\frac{1500}{150}$$

Whakawehe 1500 ki te 150.

$$t=10$$

Nā, me whakaoti te whārite t=\frac{225±1275}{150} ina he tango te ±. Tango 1275 mai i 225.

$$t=-\frac{1050}{150}$$

Whakawehe -1050 ki te 150.

$$t=-7$$

Kua oti te whārite te whakatau.

$$t=10 t=-7$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$-5250+75t^{2}-225t=0$$

Me tāpiri te 5250 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$75t^{2}-225t=5250$$

Whakawehea ngā taha e rua ki te 75.

$$\frac{75t^{2}-225t}{75}=\frac{5250}{75}$$

Mā te whakawehe ki te 75 ka wetekia te whakareanga ki te 75.

$$t^{2}+\left(-\frac{225}{75}\right)t=\frac{5250}{75}$$

Whakawehe -225 ki te 75.

$$t^{2}-3t=\frac{5250}{75}$$

Whakawehe 5250 ki te 75.

$$t^{2}-3t=70$$

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

$$t^{2}-3t+\left(-\frac{3}{2}\right)^{2}=70+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-3t+\frac{9}{4}=70+\frac{9}{4}$$

Tāpiri 70 ki te \frac{9}{4}.

$$t^{2}-3t+\frac{9}{4}=\frac{289}{4}$$

Tauwehea t^{2}-3t+\frac{9}{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}.

$$\left(t-\frac{3}{2}\right)^{2}=\frac{289}{4}$$

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

$$\sqrt{\left(t-\frac{3}{2}\right)^{2}}=\sqrt{\frac{289}{4}}$$

Whakarūnātia.

$$t-\frac{3}{2}=\frac{17}{2} t-\frac{3}{2}=-\frac{17}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$t=10 t=-7$$