Whakaoti i te 12x^2+36x+27=0

$$12 { x }^{ 2 } +36x+27=0$$

Whakaoti mō x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 3.

$$4x^{2}+12x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=12 ab=4\times 9=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Tātaihia te tapeke mō ia takirua.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

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

$$a=6 b=6$$

Tuhia anō te 4x^{2}+12x+9 hei \left(4x^{2}+6x\right)+\left(6x+9\right).

$$\left(4x^{2}+6x\right)+\left(6x+9\right)$$

Tauwehea te 2x i te tuatahi me te 3 i te rōpū tuarua.

$$2x\left(2x+3\right)+3\left(2x+3\right)$$

Whakatauwehea atu te kīanga pātahi 2x+3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x+3\right)\left(2x+3\right)$$

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te 2x+3=0.

$$x=-\frac{3}{2}$$

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.

$$12x^{2}+36x+27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 12 mō a, 36 mō b, me 27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\times 12\times 27}}{2\times 12}$$

Pūrua 36.

$$x=\frac{-36±\sqrt{1296-4\times 12\times 27}}{2\times 12}$$

Whakareatia -4 ki te 12.

$$x=\frac{-36±\sqrt{1296-48\times 27}}{2\times 12}$$

Whakareatia -48 ki te 27.

$$x=\frac{-36±\sqrt{1296-1296}}{2\times 12}$$

Tāpiri 1296 ki te -1296.

$$x=\frac{-36±\sqrt{0}}{2\times 12}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{36}{2\times 12}$$

Whakareatia 2 ki te 12.

$$x=-\frac{36}{24}$$

Whakahekea te hautanga \frac{-36}{24} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

$$x=-\frac{3}{2}$$

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.

$$12x^{2}+36x+27=0$$

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

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

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

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

Whakawehea ngā taha e rua ki te 12.

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

Mā te whakawehe ki te 12 ka wetekia te whakareanga ki te 12.

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

Whakawehe 36 ki te 12.

$$x^{2}+3x=-\frac{27}{12}$$

Whakahekea te hautanga \frac{-27}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+3x=-\frac{9}{4}$$

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

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=-\frac{9}{4}+\left(\frac{3}{2}\right)^{2}$$

Pūruatia \frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+3x+\frac{9}{4}=\frac{-9+9}{4}$$

Tāpiri -\frac{9}{4} ki te \frac{9}{4} 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.

$$x^{2}+3x+\frac{9}{4}=0$$

Tauwehea x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=0$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x+\frac{3}{2}=0 x+\frac{3}{2}=0$$

Me tango \frac{3}{2} mai i ngā taha e rua o te whārite.

$$x=-\frac{3}{2} x=-\frac{3}{2}$$

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

$$x=-\frac{3}{2}$$