Whakaoti i te y^2-10y+16=0

$$y ^ { 2 } - 10 y + 16 = 0$$

Whakaoti mō y

$$y=2$$

$$y=8$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te y^{2}-10y+16 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.

$$a+b=-10 ab=16$$

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 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 -10.

$$a=-8 b=-2$$

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

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

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

$$y=8 y=2$$

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 y^{2}+ay+by+16. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-10 ab=1\times 16=16$$

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 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 -10.

$$a=-8 b=-2$$

Tuhia anō te y^{2}-10y+16 hei \left(y^{2}-8y\right)+\left(-2y+16\right).

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

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

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

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

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

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

$$y=8 y=2$$

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.

$$y^{2}-10y+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -10 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 16}}{2}$$

Pūrua -10.

$$y=\frac{-\left(-10\right)±\sqrt{100-4\times 16}}{2}$$

Whakareatia -4 ki te 16.

$$y=\frac{-\left(-10\right)±\sqrt{100-64}}{2}$$

Tāpiri 100 ki te -64.

$$y=\frac{-\left(-10\right)±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$y=\frac{-\left(-10\right)±6}{2}$$

Ko te tauaro o -10 ko 10.

$$y=\frac{10±6}{2}$$

Nā, me whakaoti te whārite y=\frac{10±6}{2} ina he tāpiri te ±. Tāpiri 10 ki te 6.

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

Whakawehe 16 ki te 2.

$$y=8$$

Nā, me whakaoti te whārite y=\frac{10±6}{2} ina he tango te ±. Tango 6 mai i 10.

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

Whakawehe 4 ki te 2.

$$y=2$$

Kua oti te whārite te whakatau.

$$y=8 y=2$$

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.

$$y^{2}-10y+16=0$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$y^{2}-10y+16-16=-16$$

Mā te tango i te 16 i a ia ake anō ka toe ko te 0.

$$y^{2}-10y=-16$$

Whakawehea te -10, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -5. Nā, tāpiria te pūrua o te -5 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$y^{2}-10y+\left(-5\right)^{2}=-16+\left(-5\right)^{2}$$

Pūrua -5.

$$y^{2}-10y+25=-16+25$$

Tāpiri -16 ki te 25.

$$y^{2}-10y+25=9$$

Tauwehea te y^{2}-10y+25. 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}.

$$\left(y-5\right)^{2}=9$$

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

$$\sqrt{\left(y-5\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$y-5=3 y-5=-3$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$y=8 y=2$$