Whakaoti i te x^2+6/3-7/2=x+15/2

$$\frac{ { x }^{ 2 } +6 }{ 3 } - \frac{ 7 }{ 2 } = \frac{ x+15 }{ 2 }$$

Whakaoti mō x

$$x = -\frac{9}{2} = -4\frac{1}{2} = -4.5$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}+6.

$$2x^{2}+12-21=3\left(x+15\right)$$

Tangohia te 21 i te 12, ka -9.

$$2x^{2}-9=3\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+15.

$$2x^{2}-9=3x+45$$

Tangohia te 3x mai i ngā taha e rua.

$$2x^{2}-9-3x=45$$

Tangohia te 45 mai i ngā taha e rua.

$$2x^{2}-9-3x-45=0$$

Tangohia te 45 i te -9, ka -54.

$$2x^{2}-54-3x=0$$

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

$$2x^{2}-3x-54=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2x^{2}+ax+bx-54. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=2\left(-54\right)=-108$$

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 -108.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-12 b=9$$

Tuhia anō te 2x^{2}-3x-54 hei \left(2x^{2}-12x\right)+\left(9x-54\right).

$$\left(2x^{2}-12x\right)+\left(9x-54\right)$$

Tauwehea te 2x i te tuatahi me te 9 i te rōpū tuarua.

$$2x\left(x-6\right)+9\left(x-6\right)$$

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

$$\left(x-6\right)\left(2x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te 2x+9=0.

$$x=6 x=-\frac{9}{2}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}+6.

$$2x^{2}+12-21=3\left(x+15\right)$$

Tangohia te 21 i te 12, ka -9.

$$2x^{2}-9=3\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+15.

$$2x^{2}-9=3x+45$$

Tangohia te 3x mai i ngā taha e rua.

$$2x^{2}-9-3x=45$$

Tangohia te 45 mai i ngā taha e rua.

$$2x^{2}-9-3x-45=0$$

Tangohia te 45 i te -9, ka -54.

$$2x^{2}-54-3x=0$$

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.

$$2x^{2}-3x-54=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -3 mō b, me -54 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\times 2\left(-54\right)}}{2\times 2}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\times 2\left(-54\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-54\right)}}{2\times 2}$$

Whakareatia -8 ki te -54.

$$x=\frac{-\left(-3\right)±\sqrt{9+432}}{2\times 2}$$

Tāpiri 9 ki te 432.

$$x=\frac{-\left(-3\right)±\sqrt{441}}{2\times 2}$$

Tuhia te pūtakerua o te 441.

$$x=\frac{-\left(-3\right)±21}{2\times 2}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±21}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{3±21}{4}$$

Nā, me whakaoti te whārite x=\frac{3±21}{4} ina he tāpiri te ±. Tāpiri 3 ki te 21.

$$x=\frac{24}{4}$$

Whakawehe 24 ki te 4.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{3±21}{4} ina he tango te ±. Tango 21 mai i 3.

$$x=-\frac{18}{4}$$

Whakahekea te hautanga \frac{-18}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{9}{2}$$

Kua oti te whārite te whakatau.

$$x=6 x=-\frac{9}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 6, arā, te tauraro pātahi he tino iti rawa te kitea o 3,2.

$$2\left(x^{2}+6\right)-21=3\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x^{2}+6.

$$2x^{2}+12-21=3\left(x+15\right)$$

Tangohia te 21 i te 12, ka -9.

$$2x^{2}-9=3\left(x+15\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 3 ki te x+15.

$$2x^{2}-9=3x+45$$

Tangohia te 3x mai i ngā taha e rua.

$$2x^{2}-9-3x=45$$

Me tāpiri te 9 ki ngā taha e rua.

$$2x^{2}-3x=45+9$$

Tāpirihia te 45 ki te 9, ka 54.

$$2x^{2}-3x=54$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-3x}{2}=\frac{54}{2}$$

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

$$x^{2}-\frac{3}{2}x=\frac{54}{2}$$

Whakawehe 54 ki te 2.

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

Whakawehea te -\frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{4}. Nā, tāpiria te pūrua o te -\frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=27+\left(-\frac{3}{4}\right)^{2}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=27+\frac{9}{16}$$

Tāpiri 27 ki te \frac{9}{16}.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{441}{16}$$

Tauwehea x^{2}-\frac{3}{2}x+\frac{9}{16}. 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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{441}{16}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{441}{16}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{21}{4} x-\frac{3}{4}=-\frac{21}{4}$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

$$x=6 x=-\frac{9}{2}$$