Whakaoti i te x^2+12x+27=0

$$x ^ { 2 } + 12 x + 27 = 0$$

Whakaoti mō x

$$x=-9$$

$$x=-3$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+12x+27 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=27$$

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 27.

$$1,27 3,9$$

Tātaihia te tapeke mō ia takirua.

$$1+27=28 3+9=12$$

Ko te otinga te takirua ka hoatu i te tapeke 12.

$$a=3 b=9$$

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

$$\left(x+3\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x+3=0 me te x+9=0.

$$x=-3 x=-9$$

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 x^{2}+ax+bx+27. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=1\times 27=27$$

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 27.

$$1,27 3,9$$

Tātaihia te tapeke mō ia takirua.

$$1+27=28 3+9=12$$

Ko te otinga te takirua ka hoatu i te tapeke 12.

$$a=3 b=9$$

Tuhia anō te x^{2}+12x+27 hei \left(x^{2}+3x\right)+\left(9x+27\right).

$$\left(x^{2}+3x\right)+\left(9x+27\right)$$

Tauwehea te x i te tuatahi me te 9 i te rōpū tuarua.

$$x\left(x+3\right)+9\left(x+3\right)$$

Whakatauwehea atu te kīanga pātahi x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x+3\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x+3=0 me te x+9=0.

$$x=-3 x=-9$$

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.

$$x^{2}+12x+27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 12 mō b, me 27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 27}}{2}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\times 27}}{2}$$

Whakareatia -4 ki te 27.

$$x=\frac{-12±\sqrt{144-108}}{2}$$

Tāpiri 144 ki te -108.

$$x=\frac{-12±\sqrt{36}}{2}$$

Tuhia te pūtakerua o te 36.

$$x=\frac{-12±6}{2}$$

Nā, me whakaoti te whārite x=\frac{-12±6}{2} ina he tāpiri te ±. Tāpiri -12 ki te 6.

$$x=-\frac{6}{2}$$

Whakawehe -6 ki te 2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{-12±6}{2} ina he tango te ±. Tango 6 mai i -12.

$$x=-\frac{18}{2}$$

Whakawehe -18 ki te 2.

$$x=-9$$

Kua oti te whārite te whakatau.

$$x=-3 x=-9$$

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.

$$x^{2}+12x+27=0$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$x^{2}+12x+27-27=-27$$

Mā te tango i te 27 i a ia ake anō ka toe ko te 0.

$$x^{2}+12x=-27$$

Whakawehea te 12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 6. Nā, tāpiria te pūrua o te 6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+12x+6^{2}=-27+6^{2}$$

Pūrua 6.

$$x^{2}+12x+36=-27+36$$

Tāpiri -27 ki te 36.

$$x^{2}+12x+36=9$$

Tauwehea te x^{2}+12x+36. 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}.

$$\left(x+6\right)^{2}=9$$

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

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x+6=3 x+6=-3$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$x=-3 x=-9$$