Whakaoti i te 15x^2-12=8x

$$15 { x }^{ 2 } -12=8x$$

Whakaoti mō x

$$x=-\frac{2}{3}\approx -0.666666667$$

$$x = \frac{6}{5} = 1\frac{1}{5} = 1.2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 8x mai i ngā taha e rua.

$$15x^{2}-12-8x=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$15x^{2}-8x-12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 15x^{2}+ax+bx-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-8 ab=15\left(-12\right)=-180$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -180.

$$1,-180 2,-90 3,-60 4,-45 5,-36 6,-30 9,-20 10,-18 12,-15$$

Tātaihia te tapeke mō ia takirua.

$$1-180=-179 2-90=-88 3-60=-57 4-45=-41 5-36=-31 6-30=-24 9-20=-11 10-18=-8 12-15=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -8.

$$a=-18 b=10$$

Tuhia anō te 15x^{2}-8x-12 hei \left(15x^{2}-18x\right)+\left(10x-12\right).

$$\left(15x^{2}-18x\right)+\left(10x-12\right)$$

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

$$3x\left(5x-6\right)+2\left(5x-6\right)$$

Whakatauwehea atu te kīanga pātahi 5x-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-6\right)\left(3x+2\right)$$

Hei kimi otinga whārite, me whakaoti te 5x-6=0 me te 3x+2=0.

$$x=\frac{6}{5} x=-\frac{2}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 8x mai i ngā taha e rua.

$$15x^{2}-12-8x=0$$

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.

$$15x^{2}-8x-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 15 mō a, -8 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 15\left(-12\right)}}{2\times 15}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 15\left(-12\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-\left(-8\right)±\sqrt{64-60\left(-12\right)}}{2\times 15}$$

Whakareatia -60 ki te -12.

$$x=\frac{-\left(-8\right)±\sqrt{64+720}}{2\times 15}$$

Tāpiri 64 ki te 720.

$$x=\frac{-\left(-8\right)±\sqrt{784}}{2\times 15}$$

Tuhia te pūtakerua o te 784.

$$x=\frac{-\left(-8\right)±28}{2\times 15}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±28}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{8±28}{30}$$

Nā, me whakaoti te whārite x=\frac{8±28}{30} ina he tāpiri te ±. Tāpiri 8 ki te 28.

$$x=\frac{36}{30}$$

Whakahekea te hautanga \frac{36}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{6}{5}$$

Nā, me whakaoti te whārite x=\frac{8±28}{30} ina he tango te ±. Tango 28 mai i 8.

$$x=-\frac{20}{30}$$

Whakahekea te hautanga \frac{-20}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

Kua oti te whārite te whakatau.

$$x=\frac{6}{5} x=-\frac{2}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 8x mai i ngā taha e rua.

$$15x^{2}-12-8x=0$$

Me tāpiri te 12 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$15x^{2}-8x=12$$

Whakawehea ngā taha e rua ki te 15.

$$\frac{15x^{2}-8x}{15}=\frac{12}{15}$$

Mā te whakawehe ki te 15 ka wetekia te whakareanga ki te 15.

$$x^{2}-\frac{8}{15}x=\frac{12}{15}$$

Whakahekea te hautanga \frac{12}{15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}-\frac{8}{15}x=\frac{4}{5}$$

Whakawehea te -\frac{8}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{4}{15}. Nā, tāpiria te pūrua o te -\frac{4}{15} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{8}{15}x+\left(-\frac{4}{15}\right)^{2}=\frac{4}{5}+\left(-\frac{4}{15}\right)^{2}$$

Pūruatia -\frac{4}{15} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{4}{5}+\frac{16}{225}$$

Tāpiri \frac{4}{5} ki te \frac{16}{225} 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.

$$x^{2}-\frac{8}{15}x+\frac{16}{225}=\frac{196}{225}$$

Tauwehea x^{2}-\frac{8}{15}x+\frac{16}{225}. 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}.

$$\left(x-\frac{4}{15}\right)^{2}=\frac{196}{225}$$

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

$$\sqrt{\left(x-\frac{4}{15}\right)^{2}}=\sqrt{\frac{196}{225}}$$

Whakarūnātia.

$$x-\frac{4}{15}=\frac{14}{15} x-\frac{4}{15}=-\frac{14}{15}$$

Me tāpiri \frac{4}{15} ki ngā taha e rua o te whārite.

$$x=\frac{6}{5} x=-\frac{2}{3}$$