Whakaoti i te x=8/x+1/3

$$x = \frac { 8 } { x } + \frac { 1 } { 3 }$$

Whakaoti mō x

$$x = -\frac{8}{3} = -2\frac{2}{3} \approx -2.666666667$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o x me 3 ko 3x. Whakareatia \frac{8}{x} ki te \frac{3}{3}. Whakareatia \frac{1}{3} ki te \frac{x}{x}.

$$x=\frac{8\times 3}{3x}+\frac{x}{3x}$$

Tā te mea he rite te tauraro o \frac{8\times 3}{3x} me \frac{x}{3x}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$x=\frac{8\times 3+x}{3x}$$

Mahia ngā whakarea i roto o 8\times 3+x.

$$x=\frac{24+x}{3x}$$

Tangohia te \frac{24+x}{3x} mai i ngā taha e rua.

$$x-\frac{24+x}{3x}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{3x}{3x}.

$$\frac{x\times 3x}{3x}-\frac{24+x}{3x}=0$$

Tā te mea he rite te tauraro o \frac{x\times 3x}{3x} me \frac{24+x}{3x}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x\times 3x-\left(24+x\right)}{3x}=0$$

Mahia ngā whakarea i roto o x\times 3x-\left(24+x\right).

$$\frac{3x^{2}-24-x}{3x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x.

$$3x^{2}-24-x=0$$

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

$$3x^{2}-x-24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 3x^{2}+ax+bx-24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=3\left(-24\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-9 b=8$$

Tuhia anō te 3x^{2}-x-24 hei \left(3x^{2}-9x\right)+\left(8x-24\right).

$$\left(3x^{2}-9x\right)+\left(8x-24\right)$$

Tauwehea te 3x i te tuatahi me te 8 i te rōpū tuarua.

$$3x\left(x-3\right)+8\left(x-3\right)$$

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

$$\left(x-3\right)\left(3x+8\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te 3x+8=0.

$$x=3 x=-\frac{8}{3}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o x me 3 ko 3x. Whakareatia \frac{8}{x} ki te \frac{3}{3}. Whakareatia \frac{1}{3} ki te \frac{x}{x}.

$$x=\frac{8\times 3}{3x}+\frac{x}{3x}$$

Tā te mea he rite te tauraro o \frac{8\times 3}{3x} me \frac{x}{3x}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$x=\frac{8\times 3+x}{3x}$$

Mahia ngā whakarea i roto o 8\times 3+x.

$$x=\frac{24+x}{3x}$$

Tangohia te \frac{24+x}{3x} mai i ngā taha e rua.

$$x-\frac{24+x}{3x}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{3x}{3x}.

$$\frac{x\times 3x}{3x}-\frac{24+x}{3x}=0$$

Tā te mea he rite te tauraro o \frac{x\times 3x}{3x} me \frac{24+x}{3x}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x\times 3x-\left(24+x\right)}{3x}=0$$

Mahia ngā whakarea i roto o x\times 3x-\left(24+x\right).

$$\frac{3x^{2}-24-x}{3x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x.

$$3x^{2}-24-x=0$$

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.

$$3x^{2}-x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -1 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 3\left(-24\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-1\right)±\sqrt{1-12\left(-24\right)}}{2\times 3}$$

Whakareatia -12 ki te -24.

$$x=\frac{-\left(-1\right)±\sqrt{1+288}}{2\times 3}$$

Tāpiri 1 ki te 288.

$$x=\frac{-\left(-1\right)±\sqrt{289}}{2\times 3}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-\left(-1\right)±17}{2\times 3}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±17}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{1±17}{6}$$

Nā, me whakaoti te whārite x=\frac{1±17}{6} ina he tāpiri te ±. Tāpiri 1 ki te 17.

$$x=\frac{18}{6}$$

Whakawehe 18 ki te 6.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{1±17}{6} ina he tango te ±. Tango 17 mai i 1.

$$x=-\frac{16}{6}$$

Whakahekea te hautanga \frac{-16}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{8}{3}$$

Kua oti te whārite te whakatau.

$$x=3 x=-\frac{8}{3}$$

Ngā Upane mō te Whakaoti i te Pūrua

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Ko te taurea pātahi iti rawa o x me 3 ko 3x. Whakareatia \frac{8}{x} ki te \frac{3}{3}. Whakareatia \frac{1}{3} ki te \frac{x}{x}.

$$x=\frac{8\times 3}{3x}+\frac{x}{3x}$$

Tā te mea he rite te tauraro o \frac{8\times 3}{3x} me \frac{x}{3x}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$x=\frac{8\times 3+x}{3x}$$

Mahia ngā whakarea i roto o 8\times 3+x.

$$x=\frac{24+x}{3x}$$

Tangohia te \frac{24+x}{3x} mai i ngā taha e rua.

$$x-\frac{24+x}{3x}=0$$

Hei tāpiri, hei tango kīanga rānei, me whakaroha ērā kia rite ā rātou tauraro. Whakareatia x ki te \frac{3x}{3x}.

$$\frac{x\times 3x}{3x}-\frac{24+x}{3x}=0$$

Tā te mea he rite te tauraro o \frac{x\times 3x}{3x} me \frac{24+x}{3x}, me tango rāua mā te tango i ō raua taurunga.

$$\frac{x\times 3x-\left(24+x\right)}{3x}=0$$

Mahia ngā whakarea i roto o x\times 3x-\left(24+x\right).

$$\frac{3x^{2}-24-x}{3x}=0$$

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te 3x.

$$3x^{2}-24-x=0$$

Me tāpiri te 24 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$3x^{2}-x=24$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}-x}{3}=\frac{24}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}-\frac{1}{3}x=\frac{24}{3}$$

Whakawehe 24 ki te 3.

$$x^{2}-\frac{1}{3}x=8$$

Whakawehea te -\frac{1}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{6}. Nā, tāpiria te pūrua o te -\frac{1}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{3}x+\left(-\frac{1}{6}\right)^{2}=8+\left(-\frac{1}{6}\right)^{2}$$

Pūruatia -\frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=8+\frac{1}{36}$$

Tāpiri 8 ki te \frac{1}{36}.

$$x^{2}-\frac{1}{3}x+\frac{1}{36}=\frac{289}{36}$$

Tauwehea te x^{2}-\frac{1}{3}x+\frac{1}{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}.

$$\left(x-\frac{1}{6}\right)^{2}=\frac{289}{36}$$

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

$$\sqrt{\left(x-\frac{1}{6}\right)^{2}}=\sqrt{\frac{289}{36}}$$

Whakarūnātia.

$$x-\frac{1}{6}=\frac{17}{6} x-\frac{1}{6}=-\frac{17}{6}$$

Me tāpiri \frac{1}{6} ki ngā taha e rua o te whārite.

$$x=3 x=-\frac{8}{3}$$