Whakaoti i te 2a^2-12a+18=0

$$2 a ^ { 2 } - 12 a + 18 = 0$$

Whakaoti mō a

$$a=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

$$a^{2}-6a+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei a^{2}+aa+ba+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 9=9$$

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 9.

$$-1,-9 -3,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-3 b=-3$$

Tuhia anō te a^{2}-6a+9 hei \left(a^{2}-3a\right)+\left(-3a+9\right).

$$\left(a^{2}-3a\right)+\left(-3a+9\right)$$

Tauwehea te a i te tuatahi me te -3 i te rōpū tuarua.

$$a\left(a-3\right)-3\left(a-3\right)$$

Whakatauwehea atu te kīanga pātahi a-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(a-3\right)\left(a-3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(a-3\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te a-3=0.

$$a=3$$

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.

$$2a^{2}-12a+18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -12 mō b, me 18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 2\times 18}}{2\times 2}$$

Pūrua -12.

$$a=\frac{-\left(-12\right)±\sqrt{144-4\times 2\times 18}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$a=\frac{-\left(-12\right)±\sqrt{144-8\times 18}}{2\times 2}$$

Whakareatia -8 ki te 18.

$$a=\frac{-\left(-12\right)±\sqrt{144-144}}{2\times 2}$$

Tāpiri 144 ki te -144.

$$a=\frac{-\left(-12\right)±\sqrt{0}}{2\times 2}$$

Tuhia te pūtakerua o te 0.

$$a=-\frac{-12}{2\times 2}$$

Ko te tauaro o -12 ko 12.

$$a=\frac{12}{2\times 2}$$

Whakareatia 2 ki te 2.

$$a=\frac{12}{4}$$

Whakawehe 12 ki te 4.

$$a=3$$

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.

$$2a^{2}-12a+18=0$$

Me tango 18 mai i ngā taha e rua o te whārite.

$$2a^{2}-12a+18-18=-18$$

Mā te tango i te 18 i a ia ake anō ka toe ko te 0.

$$2a^{2}-12a=-18$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2a^{2}-12a}{2}=-\frac{18}{2}$$

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

$$a^{2}+\left(-\frac{12}{2}\right)a=-\frac{18}{2}$$

Whakawehe -12 ki te 2.

$$a^{2}-6a=-\frac{18}{2}$$

Whakawehe -18 ki te 2.

$$a^{2}-6a=-9$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}-6a+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Pūrua -3.

$$a^{2}-6a+9=-9+9$$

Tāpiri -9 ki te 9.

$$a^{2}-6a+9=0$$

Tauwehea a^{2}-6a+9. 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}.

$$\left(a-3\right)^{2}=0$$

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

$$\sqrt{\left(a-3\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$a-3=0 a-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$a=3 a=3$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$a=3$$