Whakaoti i te 17=22t-5t^2

$$17 = 22 t - 5 t ^ { 2 }$$

Whakaoti mō t

$$t=1$$

$$t = \frac{17}{5} = 3\frac{2}{5} = 3.4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$22t-5t^{2}=17$$

Tangohia te 17 mai i ngā taha e rua.

$$22t-5t^{2}-17=0$$

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

$$-5t^{2}+22t-17=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -5t^{2}+at+bt-17. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=22 ab=-5\left(-17\right)=85$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 85.

$$1,85 5,17$$

Tātaihia te tapeke mō ia takirua.

$$1+85=86 5+17=22$$

Ko te otinga te takirua ka hoatu i te tapeke 22.

$$a=17 b=5$$

Tuhia anō te -5t^{2}+22t-17 hei \left(-5t^{2}+17t\right)+\left(5t-17\right).

$$\left(-5t^{2}+17t\right)+\left(5t-17\right)$$

Whakatauwehea atu -t i te -5t^{2}+17t.

$$-t\left(5t-17\right)+5t-17$$

Whakatauwehea atu te kīanga pātahi 5t-17 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5t-17\right)\left(-t+1\right)$$

Hei kimi otinga whārite, me whakaoti te 5t-17=0 me te -t+1=0.

$$t=\frac{17}{5} t=1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$22t-5t^{2}=17$$

Tangohia te 17 mai i ngā taha e rua.

$$22t-5t^{2}-17=0$$

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.

$$-5t^{2}+22t-17=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, 22 mō b, me -17 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$t=\frac{-22±\sqrt{22^{2}-4\left(-5\right)\left(-17\right)}}{2\left(-5\right)}$$

Pūrua 22.

$$t=\frac{-22±\sqrt{484-4\left(-5\right)\left(-17\right)}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$t=\frac{-22±\sqrt{484+20\left(-17\right)}}{2\left(-5\right)}$$

Whakareatia 20 ki te -17.

$$t=\frac{-22±\sqrt{484-340}}{2\left(-5\right)}$$

Tāpiri 484 ki te -340.

$$t=\frac{-22±\sqrt{144}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te 144.

$$t=\frac{-22±12}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$t=\frac{-22±12}{-10}$$

Nā, me whakaoti te whārite t=\frac{-22±12}{-10} ina he tāpiri te ±. Tāpiri -22 ki te 12.

$$t=-\frac{10}{-10}$$

Whakawehe -10 ki te -10.

$$t=1$$

Nā, me whakaoti te whārite t=\frac{-22±12}{-10} ina he tango te ±. Tango 12 mai i -22.

$$t=-\frac{34}{-10}$$

Whakahekea te hautanga \frac{-34}{-10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$t=\frac{17}{5}$$

Kua oti te whārite te whakatau.

$$t=1 t=\frac{17}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$22t-5t^{2}=17$$

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.

$$-5t^{2}+22t=17$$

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5t^{2}+22t}{-5}=\frac{17}{-5}$$

Mā te whakawehe ki te -5 ka wetekia te whakareanga ki te -5.

$$t^{2}+\frac{22}{-5}t=\frac{17}{-5}$$

Whakawehe 22 ki te -5.

$$t^{2}-\frac{22}{5}t=\frac{17}{-5}$$

Whakawehe 17 ki te -5.

$$t^{2}-\frac{22}{5}t=-\frac{17}{5}$$

Whakawehea te -\frac{22}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{5}. Nā, tāpiria te pūrua o te -\frac{11}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$t^{2}-\frac{22}{5}t+\left(-\frac{11}{5}\right)^{2}=-\frac{17}{5}+\left(-\frac{11}{5}\right)^{2}$$

Pūruatia -\frac{11}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=-\frac{17}{5}+\frac{121}{25}$$

Tāpiri -\frac{17}{5} ki te \frac{121}{25} 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.

$$t^{2}-\frac{22}{5}t+\frac{121}{25}=\frac{36}{25}$$

Tauwehea t^{2}-\frac{22}{5}t+\frac{121}{25}. 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}.

$$\left(t-\frac{11}{5}\right)^{2}=\frac{36}{25}$$

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

$$\sqrt{\left(t-\frac{11}{5}\right)^{2}}=\sqrt{\frac{36}{25}}$$

Whakarūnātia.

$$t-\frac{11}{5}=\frac{6}{5} t-\frac{11}{5}=-\frac{6}{5}$$

Me tāpiri \frac{11}{5} ki ngā taha e rua o te whārite.

$$t=\frac{17}{5} t=1$$