Whakaoti i te x+6/x+2=x

$$\frac { x + 6 } { x + 2 } = x$$

Whakaoti mō x

$$x=-3$$

$$x=2$$

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. Whakareatia ngā taha e rua o te whārite ki te x+2.

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

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

$$x+6=x^{2}+2x$$

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

$$x+6-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$x+6-x^{2}-2x=0$$

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

$$-x+6-x^{2}=0$$

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

$$-x^{2}-x+6=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=-6=-6$$

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 -6.

$$1,-6 2,-3$$

Tātaihia te tapeke mō ia takirua.

$$1-6=-5 2-3=-1$$

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

$$a=2 b=-3$$

Tuhia anō te -x^{2}-x+6 hei \left(-x^{2}+2x\right)+\left(-3x+6\right).

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

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

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

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

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

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

$$x=2 x=-3$$

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. Whakareatia ngā taha e rua o te whārite ki te x+2.

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

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

$$x+6=x^{2}+2x$$

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

$$x+6-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$x+6-x^{2}-2x=0$$

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

$$-x+6-x^{2}=0$$

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.

$$-x^{2}-x+6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -1 mō b, me 6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)\times 6}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-1\right)±\sqrt{1+4\times 6}}{2\left(-1\right)}$$

Whakareatia 4 ki te 6.

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

Tāpiri 1 ki te 24.

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

Tuhia te pūtakerua o te 25.

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

Ko te tauaro o -1 ko 1.

$$x=\frac{1±5}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{1±5}{-2}$$

Nā, me whakaoti te whārite x=\frac{1±5}{-2} ina he tāpiri te ±. Tāpiri 1 ki te 5.

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

Whakawehe 6 ki te -2.

$$x=-3$$

Nā, me whakaoti te whārite x=\frac{1±5}{-2} ina he tango te ±. Tango 5 mai i 1.

$$x=-\frac{4}{-2}$$

Whakawehe -4 ki te -2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=-3 x=2$$

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. Whakareatia ngā taha e rua o te whārite ki te x+2.

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

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

$$x+6=x^{2}+2x$$

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

$$x+6-x^{2}=2x$$

Tangohia te 2x mai i ngā taha e rua.

$$x+6-x^{2}-2x=0$$

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

$$-x+6-x^{2}=0$$

Tangohia te 6 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-x-x^{2}=-6$$

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.

$$-x^{2}-x=-6$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-x}{-1}=-\frac{6}{-1}$$

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

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

Whakawehe -1 ki te -1.

$$x^{2}+x=-\frac{6}{-1}$$

Whakawehe -6 ki te -1.

$$x^{2}+x=6$$

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

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

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+x+\frac{1}{4}=6+\frac{1}{4}$$

Tāpiri 6 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{25}{4}$$

Tauwehea x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$x+\frac{1}{2}=\frac{5}{2} x+\frac{1}{2}=-\frac{5}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$x=2 x=-3$$