Whakaoti i te 2y^2-7y+5=0

$$2 y ^ { 2 } - 7 y + 5 = 0$$

Whakaoti mō y

$$y=1$$

$$y = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

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 2y^{2}+ay+by+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-7 ab=2\times 5=10$$

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 10.

$$-1,-10 -2,-5$$

Tātaihia te tapeke mō ia takirua.

$$-1-10=-11 -2-5=-7$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-5 b=-2$$

Tuhia anō te 2y^{2}-7y+5 hei \left(2y^{2}-5y\right)+\left(-2y+5\right).

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

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

$$y\left(2y-5\right)-\left(2y-5\right)$$

Whakatauwehea atu te kīanga pātahi 2y-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2y-5\right)\left(y-1\right)$$

Hei kimi otinga whārite, me whakaoti te 2y-5=0 me te y-1=0.

$$y=\frac{5}{2} y=1$$

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.

$$2y^{2}-7y+5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -7 mō b, me 5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 2\times 5}}{2\times 2}$$

Pūrua -7.

$$y=\frac{-\left(-7\right)±\sqrt{49-4\times 2\times 5}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$y=\frac{-\left(-7\right)±\sqrt{49-8\times 5}}{2\times 2}$$

Whakareatia -8 ki te 5.

$$y=\frac{-\left(-7\right)±\sqrt{49-40}}{2\times 2}$$

Tāpiri 49 ki te -40.

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

Tuhia te pūtakerua o te 9.

$$y=\frac{-\left(-7\right)±3}{2\times 2}$$

Ko te tauaro o -7 ko 7.

$$y=\frac{7±3}{2\times 2}$$

Whakareatia 2 ki te 2.

$$y=\frac{7±3}{4}$$

Nā, me whakaoti te whārite y=\frac{7±3}{4} ina he tāpiri te ±. Tāpiri 7 ki te 3.

$$y=\frac{10}{4}$$

Whakahekea te hautanga \frac{10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=\frac{5}{2}$$

Nā, me whakaoti te whārite y=\frac{7±3}{4} ina he tango te ±. Tango 3 mai i 7.

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

Whakawehe 4 ki te 4.

$$y=1$$

Kua oti te whārite te whakatau.

$$y=\frac{5}{2} y=1$$

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.

$$2y^{2}-7y+5=0$$

Me tango 5 mai i ngā taha e rua o te whārite.

$$2y^{2}-7y+5-5=-5$$

Mā te tango i te 5 i a ia ake anō ka toe ko te 0.

$$2y^{2}-7y=-5$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2y^{2}-7y}{2}=-\frac{5}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$y^{2}-\frac{7}{2}y=-\frac{5}{2}$$

Whakawehea te -\frac{7}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{4}. Nā, tāpiria te pūrua o te -\frac{7}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{7}{2}y+\left(-\frac{7}{4}\right)^{2}=-\frac{5}{2}+\left(-\frac{7}{4}\right)^{2}$$

Pūruatia -\frac{7}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{7}{2}y+\frac{49}{16}=-\frac{5}{2}+\frac{49}{16}$$

Tāpiri -\frac{5}{2} ki te \frac{49}{16} 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.

$$y^{2}-\frac{7}{2}y+\frac{49}{16}=\frac{9}{16}$$

Tauwehea y^{2}-\frac{7}{2}y+\frac{49}{16}. 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}.

$$\left(y-\frac{7}{4}\right)^{2}=\frac{9}{16}$$

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

$$\sqrt{\left(y-\frac{7}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Whakarūnātia.

$$y-\frac{7}{4}=\frac{3}{4} y-\frac{7}{4}=-\frac{3}{4}$$

Me tāpiri \frac{7}{4} ki ngā taha e rua o te whārite.

$$y=\frac{5}{2} y=1$$