Whakaoti i te y^2+9y=-8

$$y ^ { 2 } + 9 y = - 8$$

Whakaoti mō y

$$y=-8$$

$$y=-1$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 8 ki ngā taha e rua.

$$y^{2}+9y+8=0$$

Hei whakaoti i te whārite, whakatauwehea te y^{2}+9y+8 mā te whakamahi i te tātai y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=9 ab=8$$

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 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.

$$a=1 b=8$$

Me tuhi anō te kīanga whakatauwehe \left(y+a\right)\left(y+b\right) mā ngā uara i tātaihia.

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

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

$$y=-1 y=-8$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 8 ki ngā taha e rua.

$$y^{2}+9y+8=0$$

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

$$a+b=9 ab=1\times 8=8$$

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 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.

$$a=1 b=8$$

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

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

Tauwehea te y i te tuatahi me te 8 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi y+1 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$y=-1 y=-8$$

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.

$$y^{2}+9y=-8$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

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

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

$$y^{2}+9y-\left(-8\right)=0$$

Tango -8 mai i 0.

$$y^{2}+9y+8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 9 mō b, me 8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-9±\sqrt{9^{2}-4\times 8}}{2}$$

Pūrua 9.

$$y=\frac{-9±\sqrt{81-4\times 8}}{2}$$

Whakareatia -4 ki te 8.

$$y=\frac{-9±\sqrt{81-32}}{2}$$

Tāpiri 81 ki te -32.

$$y=\frac{-9±\sqrt{49}}{2}$$

Tuhia te pūtakerua o te 49.

$$y=\frac{-9±7}{2}$$

Nā, me whakaoti te whārite y=\frac{-9±7}{2} ina he tāpiri te ±. Tāpiri -9 ki te 7.

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

Whakawehe -2 ki te 2.

$$y=-1$$

Nā, me whakaoti te whārite y=\frac{-9±7}{2} ina he tango te ±. Tango 7 mai i -9.

$$y=-\frac{16}{2}$$

Whakawehe -16 ki te 2.

$$y=-8$$

Kua oti te whārite te whakatau.

$$y=-1 y=-8$$

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.

$$y^{2}+9y=-8$$

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

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

Pūruatia \frac{9}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$y^{2}+9y+\frac{81}{4}=-8+\frac{81}{4}$$

Tāpiri -8 ki te \frac{81}{4}.

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

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

$$\left(y+\frac{9}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(y+\frac{9}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

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

Me tango \frac{9}{2} mai i ngā taha e rua o te whārite.

$$y=-1 y=-8$$