Whakaoti i te 25y^2-54y-63=0

$$25 y ^ { 2 } - 54 y - 63 = 0$$

Whakaoti mō y

$$y=-\frac{21}{25}=-0.84$$

$$y=3$$

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 25y^{2}+ay+by-63. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-54 ab=25\left(-63\right)=-1575$$

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 -1575.

$$1,-1575 3,-525 5,-315 7,-225 9,-175 15,-105 21,-75 25,-63 35,-45$$

Tātaihia te tapeke mō ia takirua.

$$1-1575=-1574 3-525=-522 5-315=-310 7-225=-218 9-175=-166 15-105=-90 21-75=-54 25-63=-38 35-45=-10$$

Ko te otinga te takirua ka hoatu i te tapeke -54.

$$a=-75 b=21$$

Tuhia anō te 25y^{2}-54y-63 hei \left(25y^{2}-75y\right)+\left(21y-63\right).

$$\left(25y^{2}-75y\right)+\left(21y-63\right)$$

Tauwehea te 25y i te tuatahi me te 21 i te rōpū tuarua.

$$25y\left(y-3\right)+21\left(y-3\right)$$

Whakatauwehea atu te kīanga pātahi y-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-3\right)\left(25y+21\right)$$

Hei kimi otinga whārite, me whakaoti te y-3=0 me te 25y+21=0.

$$y=3 y=-\frac{21}{25}$$

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.

$$25y^{2}-54y-63=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 25 mō a, -54 mō b, me -63 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-54\right)±\sqrt{\left(-54\right)^{2}-4\times 25\left(-63\right)}}{2\times 25}$$

Pūrua -54.

$$y=\frac{-\left(-54\right)±\sqrt{2916-4\times 25\left(-63\right)}}{2\times 25}$$

Whakareatia -4 ki te 25.

$$y=\frac{-\left(-54\right)±\sqrt{2916-100\left(-63\right)}}{2\times 25}$$

Whakareatia -100 ki te -63.

$$y=\frac{-\left(-54\right)±\sqrt{2916+6300}}{2\times 25}$$

Tāpiri 2916 ki te 6300.

$$y=\frac{-\left(-54\right)±\sqrt{9216}}{2\times 25}$$

Tuhia te pūtakerua o te 9216.

$$y=\frac{-\left(-54\right)±96}{2\times 25}$$

Ko te tauaro o -54 ko 54.

$$y=\frac{54±96}{2\times 25}$$

Whakareatia 2 ki te 25.

$$y=\frac{54±96}{50}$$

Nā, me whakaoti te whārite y=\frac{54±96}{50} ina he tāpiri te ±. Tāpiri 54 ki te 96.

$$y=\frac{150}{50}$$

Whakawehe 150 ki te 50.

$$y=3$$

Nā, me whakaoti te whārite y=\frac{54±96}{50} ina he tango te ±. Tango 96 mai i 54.

$$y=-\frac{42}{50}$$

Whakahekea te hautanga \frac{-42}{50} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=-\frac{21}{25}$$

Kua oti te whārite te whakatau.

$$y=3 y=-\frac{21}{25}$$

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.

$$25y^{2}-54y-63=0$$

Me tāpiri 63 ki ngā taha e rua o te whārite.

$$25y^{2}-54y-63-\left(-63\right)=-\left(-63\right)$$

Mā te tango i te -63 i a ia ake anō ka toe ko te 0.

$$25y^{2}-54y=-\left(-63\right)$$

Tango -63 mai i 0.

$$25y^{2}-54y=63$$

Whakawehea ngā taha e rua ki te 25.

$$\frac{25y^{2}-54y}{25}=\frac{63}{25}$$

Mā te whakawehe ki te 25 ka wetekia te whakareanga ki te 25.

$$y^{2}-\frac{54}{25}y=\frac{63}{25}$$

Whakawehea te -\frac{54}{25}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{27}{25}. Nā, tāpiria te pūrua o te -\frac{27}{25} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{54}{25}y+\left(-\frac{27}{25}\right)^{2}=\frac{63}{25}+\left(-\frac{27}{25}\right)^{2}$$

Pūruatia -\frac{27}{25} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{54}{25}y+\frac{729}{625}=\frac{63}{25}+\frac{729}{625}$$

Tāpiri \frac{63}{25} ki te \frac{729}{625} 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.

$$y^{2}-\frac{54}{25}y+\frac{729}{625}=\frac{2304}{625}$$

Tauwehea y^{2}-\frac{54}{25}y+\frac{729}{625}. 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}.

$$\left(y-\frac{27}{25}\right)^{2}=\frac{2304}{625}$$

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

$$\sqrt{\left(y-\frac{27}{25}\right)^{2}}=\sqrt{\frac{2304}{625}}$$

Whakarūnātia.

$$y-\frac{27}{25}=\frac{48}{25} y-\frac{27}{25}=-\frac{48}{25}$$

Me tāpiri \frac{27}{25} ki ngā taha e rua o te whārite.

$$y=3 y=-\frac{21}{25}$$