Whakaoti i te 4y^2-9y+2=0

$$4 y ^ { 2 } - 9 y + 2 = 0$$

Whakaoti mō y

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

$$y=2$$

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 4y^{2}+ay+by+2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-9 ab=4\times 2=8$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 8.

$$-1,-8 -2,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-8=-9 -2-4=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -9.

$$a=-8 b=-1$$

Tuhia anō te 4y^{2}-9y+2 hei \left(4y^{2}-8y\right)+\left(-y+2\right).

$$\left(4y^{2}-8y\right)+\left(-y+2\right)$$

Tauwehea te 4y i te tuatahi me te -1 i te rōpū tuarua.

$$4y\left(y-2\right)-\left(y-2\right)$$

Whakatauwehea atu te kīanga pātahi y-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(y-2\right)\left(4y-1\right)$$

Hei kimi otinga whārite, me whakaoti te y-2=0 me te 4y-1=0.

$$y=2 y=\frac{1}{4}$$

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.

$$4y^{2}-9y+2=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, -9 mō b, me 2 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 4\times 2}}{2\times 4}$$

Pūrua -9.

$$y=\frac{-\left(-9\right)±\sqrt{81-4\times 4\times 2}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$y=\frac{-\left(-9\right)±\sqrt{81-16\times 2}}{2\times 4}$$

Whakareatia -16 ki te 2.

$$y=\frac{-\left(-9\right)±\sqrt{81-32}}{2\times 4}$$

Tāpiri 81 ki te -32.

$$y=\frac{-\left(-9\right)±\sqrt{49}}{2\times 4}$$

Tuhia te pūtakerua o te 49.

$$y=\frac{-\left(-9\right)±7}{2\times 4}$$

Ko te tauaro o -9 ko 9.

$$y=\frac{9±7}{2\times 4}$$

Whakareatia 2 ki te 4.

$$y=\frac{9±7}{8}$$

Nā, me whakaoti te whārite y=\frac{9±7}{8} ina he tāpiri te ±. Tāpiri 9 ki te 7.

$$y=\frac{16}{8}$$

Whakawehe 16 ki te 8.

$$y=2$$

Nā, me whakaoti te whārite y=\frac{9±7}{8} ina he tango te ±. Tango 7 mai i 9.

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

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

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

Kua oti te whārite te whakatau.

$$y=2 y=\frac{1}{4}$$

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.

$$4y^{2}-9y+2=0$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$4y^{2}-9y+2-2=-2$$

Mā te tango i te 2 i a ia ake anō ka toe ko te 0.

$$4y^{2}-9y=-2$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4y^{2}-9y}{4}=-\frac{2}{4}$$

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

$$y^{2}-\frac{9}{4}y=-\frac{2}{4}$$

Whakahekea te hautanga \frac{-2}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y^{2}-\frac{9}{4}y=-\frac{1}{2}$$

Whakawehea te -\frac{9}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{8}. Nā, tāpiria te pūrua o te -\frac{9}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{9}{4}y+\left(-\frac{9}{8}\right)^{2}=-\frac{1}{2}+\left(-\frac{9}{8}\right)^{2}$$

Pūruatia -\frac{9}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{9}{4}y+\frac{81}{64}=-\frac{1}{2}+\frac{81}{64}$$

Tāpiri -\frac{1}{2} ki te \frac{81}{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.

$$y^{2}-\frac{9}{4}y+\frac{81}{64}=\frac{49}{64}$$

Tauwehea te y^{2}-\frac{9}{4}y+\frac{81}{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}.

$$\left(y-\frac{9}{8}\right)^{2}=\frac{49}{64}$$

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

$$\sqrt{\left(y-\frac{9}{8}\right)^{2}}=\sqrt{\frac{49}{64}}$$

Whakarūnātia.

$$y-\frac{9}{8}=\frac{7}{8} y-\frac{9}{8}=-\frac{7}{8}$$

Me tāpiri \frac{9}{8} ki ngā taha e rua o te whārite.

$$y=2 y=\frac{1}{4}$$