Whakaoti i te 2/x+2-1/3=x/3

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

Whakaoti mō x

$$x=-4$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki -2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+2,3.

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

Whakareatia te 3 ki te 2, ka 6.

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

Whakareatia te 3 ki te -\frac{1}{3}, ka -1.

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

Hei kimi i te tauaro o x+2, kimihia te tauaro o ia taurangi.

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

Tangohia te 2 i te 6, ka 4.

$$4-x=\left(x+2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te x.

$$4-x=x^{2}+2x$$

Tangohia te x^{2} mai i ngā taha e rua.

$$4-x-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$4-x-x^{2}-2x=0$$

Pahekotia te -x me -2x, ka -3x.

$$4-3x-x^{2}=0$$

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

$$-x^{2}-3x+4=0$$

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

$$a+b=-3 ab=-4=-4$$

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 -4.

$$1,-4 2,-2$$

Tātaihia te tapeke mō ia takirua.

$$1-4=-3 2-2=0$$

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

$$a=1 b=-4$$

Tuhia anō te -x^{2}-3x+4 hei \left(-x^{2}+x\right)+\left(-4x+4\right).

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

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(-x+1\right)+4\left(-x+1\right)$$

Whakatauwehea atu te kīanga pātahi -x+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+1\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te -x+1=0 me te x+4=0.

$$x=1 x=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki -2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+2,3.

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

Whakareatia te 3 ki te 2, ka 6.

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

Whakareatia te 3 ki te -\frac{1}{3}, ka -1.

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

Hei kimi i te tauaro o x+2, kimihia te tauaro o ia taurangi.

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

Tangohia te 2 i te 6, ka 4.

$$4-x=\left(x+2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te x.

$$4-x=x^{2}+2x$$

Tangohia te x^{2} mai i ngā taha e rua.

$$4-x-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$4-x-x^{2}-2x=0$$

Pahekotia te -x me -2x, ka -3x.

$$4-3x-x^{2}=0$$

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.

$$-x^{2}-3x+4=0$$

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

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

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-1\right)\times 4}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-3\right)±\sqrt{9+4\times 4}}{2\left(-1\right)}$$

Whakareatia 4 ki te 4.

$$x=\frac{-\left(-3\right)±\sqrt{9+16}}{2\left(-1\right)}$$

Tāpiri 9 ki te 16.

$$x=\frac{-\left(-3\right)±\sqrt{25}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-\left(-3\right)±5}{2\left(-1\right)}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±5}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{3±5}{-2}$$

Nā, me whakaoti te whārite x=\frac{3±5}{-2} ina he tāpiri te ±. Tāpiri 3 ki te 5.

$$x=\frac{8}{-2}$$

Whakawehe 8 ki te -2.

$$x=-4$$

Nā, me whakaoti te whārite x=\frac{3±5}{-2} ina he tango te ±. Tango 5 mai i 3.

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

Whakawehe -2 ki te -2.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=-4 x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki -2 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 3\left(x+2\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+2,3.

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

Whakareatia te 3 ki te 2, ka 6.

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

Whakareatia te 3 ki te -\frac{1}{3}, ka -1.

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

Hei kimi i te tauaro o x+2, kimihia te tauaro o ia taurangi.

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

Tangohia te 2 i te 6, ka 4.

$$4-x=\left(x+2\right)x$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te x.

$$4-x=x^{2}+2x$$

Tangohia te x^{2} mai i ngā taha e rua.

$$4-x-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$4-x-x^{2}-2x=0$$

Pahekotia te -x me -2x, ka -3x.

$$4-3x-x^{2}=0$$

Tangohia te 4 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-3x-x^{2}=-4$$

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.

$$-x^{2}-3x=-4$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-3x}{-1}=-\frac{4}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\left(-\frac{3}{-1}\right)x=-\frac{4}{-1}$$

Whakawehe -3 ki te -1.

$$x^{2}+3x=-\frac{4}{-1}$$

Whakawehe -4 ki te -1.

$$x^{2}+3x=4$$

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.

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

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

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

Tāpiri 4 ki te \frac{9}{4}.

$$x^{2}+3x+\frac{9}{4}=\frac{25}{4}$$

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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

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

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

$$x=1 x=-4$$