Whakaoti i te 8-26y+11y^2=0

$$8 - 26 y + 11 y ^ { 2 } = 0$$

Whakaoti mō y

$$y=2$$

$$y=\frac{4}{11}\approx 0.363636364$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$11y^{2}-26y+8=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 11y^{2}+ay+by+8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-26 ab=11\times 8=88$$

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 88.

$$-1,-88 -2,-44 -4,-22 -8,-11$$

Tātaihia te tapeke mō ia takirua.

$$-1-88=-89 -2-44=-46 -4-22=-26 -8-11=-19$$

Ko te otinga te takirua ka hoatu i te tapeke -26.

$$a=-22 b=-4$$

Tuhia anō te 11y^{2}-26y+8 hei \left(11y^{2}-22y\right)+\left(-4y+8\right).

$$\left(11y^{2}-22y\right)+\left(-4y+8\right)$$

Tauwehea te 11y i te tuatahi me te -4 i te rōpū tuarua.

$$11y\left(y-2\right)-4\left(y-2\right)$$

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

$$\left(y-2\right)\left(11y-4\right)$$

Hei kimi otinga whārite, me whakaoti te y-2=0 me te 11y-4=0.

$$y=2 y=\frac{4}{11}$$

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.

$$11y^{2}-26y+8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 11 mō a, -26 mō b, me 8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 11\times 8}}{2\times 11}$$

Pūrua -26.

$$y=\frac{-\left(-26\right)±\sqrt{676-4\times 11\times 8}}{2\times 11}$$

Whakareatia -4 ki te 11.

$$y=\frac{-\left(-26\right)±\sqrt{676-44\times 8}}{2\times 11}$$

Whakareatia -44 ki te 8.

$$y=\frac{-\left(-26\right)±\sqrt{676-352}}{2\times 11}$$

Tāpiri 676 ki te -352.

$$y=\frac{-\left(-26\right)±\sqrt{324}}{2\times 11}$$

Tuhia te pūtakerua o te 324.

$$y=\frac{-\left(-26\right)±18}{2\times 11}$$

Ko te tauaro o -26 ko 26.

$$y=\frac{26±18}{2\times 11}$$

Whakareatia 2 ki te 11.

$$y=\frac{26±18}{22}$$

Nā, me whakaoti te whārite y=\frac{26±18}{22} ina he tāpiri te ±. Tāpiri 26 ki te 18.

$$y=\frac{44}{22}$$

Whakawehe 44 ki te 22.

$$y=2$$

Nā, me whakaoti te whārite y=\frac{26±18}{22} ina he tango te ±. Tango 18 mai i 26.

$$y=\frac{8}{22}$$

Whakahekea te hautanga \frac{8}{22} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$y=\frac{4}{11}$$

Kua oti te whārite te whakatau.

$$y=2 y=\frac{4}{11}$$

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.

$$11y^{2}-26y+8=0$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$11y^{2}-26y+8-8=-8$$

Mā te tango i te 8 i a ia ake anō ka toe ko te 0.

$$11y^{2}-26y=-8$$

Whakawehea ngā taha e rua ki te 11.

$$\frac{11y^{2}-26y}{11}=-\frac{8}{11}$$

Mā te whakawehe ki te 11 ka wetekia te whakareanga ki te 11.

$$y^{2}-\frac{26}{11}y=-\frac{8}{11}$$

Whakawehea te -\frac{26}{11}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{11}. Nā, tāpiria te pūrua o te -\frac{13}{11} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-\frac{26}{11}y+\left(-\frac{13}{11}\right)^{2}=-\frac{8}{11}+\left(-\frac{13}{11}\right)^{2}$$

Pūruatia -\frac{13}{11} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}-\frac{26}{11}y+\frac{169}{121}=-\frac{8}{11}+\frac{169}{121}$$

Tāpiri -\frac{8}{11} ki te \frac{169}{121} 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.

$$y^{2}-\frac{26}{11}y+\frac{169}{121}=\frac{81}{121}$$

Tauwehea y^{2}-\frac{26}{11}y+\frac{169}{121}. 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}.

$$\left(y-\frac{13}{11}\right)^{2}=\frac{81}{121}$$

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

$$\sqrt{\left(y-\frac{13}{11}\right)^{2}}=\sqrt{\frac{81}{121}}$$

Whakarūnātia.

$$y-\frac{13}{11}=\frac{9}{11} y-\frac{13}{11}=-\frac{9}{11}$$

Me tāpiri \frac{13}{11} ki ngā taha e rua o te whārite.

$$y=2 y=\frac{4}{11}$$