Whakaoti i te 0=(2y-1)(8-y)

$$0 = ( 2 y - 1 ) ( 8 - y )$$

Whakaoti mō y

$$y=8$$

$$y=\frac{1}{2}=0.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakamahia te āhuatanga tuaritanga hei whakarea te 2y-1 ki te 8-y ka whakakotahi i ngā kupu rite.

$$0=17y-2y^{2}-8$$

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

$$17y-2y^{2}-8=0$$

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

$$-2y^{2}+17y-8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -2y^{2}+ay+by-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=-2\left(-8\right)=16$$

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 16.

$$1,16 2,8 4,4$$

Tātaihia te tapeke mō ia takirua.

$$1+16=17 2+8=10 4+4=8$$

Ko te otinga te takirua ka hoatu i te tapeke 17.

$$a=16 b=1$$

Tuhia anō te -2y^{2}+17y-8 hei \left(-2y^{2}+16y\right)+\left(y-8\right).

$$\left(-2y^{2}+16y\right)+\left(y-8\right)$$

Tauwehea te 2y i te tuatahi me te -1 i te rōpū tuarua.

$$2y\left(-y+8\right)-\left(-y+8\right)$$

Whakatauwehea atu te kīanga pātahi -y+8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-y+8\right)\left(2y-1\right)$$

Hei kimi otinga whārite, me whakaoti te -y+8=0 me te 2y-1=0.

$$y=8 y=\frac{1}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2y-1 ki te 8-y ka whakakotahi i ngā kupu rite.

$$0=17y-2y^{2}-8$$

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

$$17y-2y^{2}-8=0$$

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.

$$-2y^{2}+17y-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, 17 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-17±\sqrt{17^{2}-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}$$

Pūrua 17.

$$y=\frac{-17±\sqrt{289-4\left(-2\right)\left(-8\right)}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$y=\frac{-17±\sqrt{289+8\left(-8\right)}}{2\left(-2\right)}$$

Whakareatia 8 ki te -8.

$$y=\frac{-17±\sqrt{289-64}}{2\left(-2\right)}$$

Tāpiri 289 ki te -64.

$$y=\frac{-17±\sqrt{225}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 225.

$$y=\frac{-17±15}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$y=\frac{-17±15}{-4}$$

Nā, me whakaoti te whārite y=\frac{-17±15}{-4} ina he tāpiri te ±. Tāpiri -17 ki te 15.

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

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

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

Nā, me whakaoti te whārite y=\frac{-17±15}{-4} ina he tango te ±. Tango 15 mai i -17.

$$y=-\frac{32}{-4}$$

Whakawehe -32 ki te -4.

$$y=8$$

Kua oti te whārite te whakatau.

$$y=\frac{1}{2} y=8$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tuaritanga hei whakarea te 2y-1 ki te 8-y ka whakakotahi i ngā kupu rite.

$$0=17y-2y^{2}-8$$

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

$$17y-2y^{2}-8=0$$

Me tāpiri te 8 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$17y-2y^{2}=8$$

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.

$$-2y^{2}+17y=8$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2y^{2}+17y}{-2}=\frac{8}{-2}$$

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

$$y^{2}+\frac{17}{-2}y=\frac{8}{-2}$$

Whakawehe 17 ki te -2.

$$y^{2}-\frac{17}{2}y=\frac{8}{-2}$$

Whakawehe 8 ki te -2.

$$y^{2}-\frac{17}{2}y=-4$$

Whakawehea te -\frac{17}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{17}{4}. Nā, tāpiria te pūrua o te -\frac{17}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{17}{2}y+\left(-\frac{17}{4}\right)^{2}=-4+\left(-\frac{17}{4}\right)^{2}$$

Pūruatia -\frac{17}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{17}{2}y+\frac{289}{16}=-4+\frac{289}{16}$$

Tāpiri -4 ki te \frac{289}{16}.

$$y^{2}-\frac{17}{2}y+\frac{289}{16}=\frac{225}{16}$$

Tauwehea y^{2}-\frac{17}{2}y+\frac{289}{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}.

$$\left(y-\frac{17}{4}\right)^{2}=\frac{225}{16}$$

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

$$\sqrt{\left(y-\frac{17}{4}\right)^{2}}=\sqrt{\frac{225}{16}}$$

Whakarūnātia.

$$y-\frac{17}{4}=\frac{15}{4} y-\frac{17}{4}=-\frac{15}{4}$$

Me tāpiri \frac{17}{4} ki ngā taha e rua o te whārite.

$$y=8 y=\frac{1}{2}$$