Whakaoti i te x+1/x+x/x+1=5/2

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

Whakaoti mō x

$$x=1$$

$$x=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+1,2.

$$\left(2x+2\right)\left(x+1\right)+2xx=5x\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+2 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

Pahekotia te 2x^{2} me 2x^{2}, ka 4x^{2}.

$$4x^{2}+4x+2=5x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x+1.

$$4x^{2}+4x+2=5x^{2}+5x$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$4x^{2}+4x+2-5x^{2}=5x$$

Pahekotia te 4x^{2} me -5x^{2}, ka -x^{2}.

$$-x^{2}+4x+2=5x$$

Tangohia te 5x mai i ngā taha e rua.

$$-x^{2}+4x+2-5x=0$$

Pahekotia te 4x me -5x, ka -x.

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

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

$$a+b=-1 ab=-2=-2$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=1 b=-2$$

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

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

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

$$x\left(-x+1\right)+2\left(-x+1\right)$$

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

$$\left(-x+1\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te -x+1=0 me te x+2=0.

$$x=1 x=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+1,2.

$$\left(2x+2\right)\left(x+1\right)+2xx=5x\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+2 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

Pahekotia te 2x^{2} me 2x^{2}, ka 4x^{2}.

$$4x^{2}+4x+2=5x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x+1.

$$4x^{2}+4x+2=5x^{2}+5x$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$4x^{2}+4x+2-5x^{2}=5x$$

Pahekotia te 4x^{2} me -5x^{2}, ka -x^{2}.

$$-x^{2}+4x+2=5x$$

Tangohia te 5x mai i ngā taha e rua.

$$-x^{2}+4x+2-5x=0$$

Pahekotia te 4x me -5x, ka -x.

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

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

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

Whakareatia -4 ki te -1.

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

Whakareatia 4 ki te 2.

$$x=\frac{-\left(-1\right)±\sqrt{1+8}}{2\left(-1\right)}$$

Tāpiri 1 ki te 8.

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

Tuhia te pūtakerua o te 9.

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

Ko te tauaro o -1 ko 1.

$$x=\frac{1±3}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{1±3}{-2}$$

Nā, me whakaoti te whārite x=\frac{1±3}{-2} ina he tāpiri te ±. Tāpiri 1 ki te 3.

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

Whakawehe 4 ki te -2.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{1±3}{-2} ina he tango te ±. Tango 3 mai i 1.

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

Whakawehe -2 ki te -2.

$$x=1$$

Kua oti te whārite te whakatau.

$$x=-2 x=1$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -1,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 2x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+1,2.

$$\left(2x+2\right)\left(x+1\right)+2xx=5x\left(x+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 2x+2 ki te x+1 ka whakakotahi i ngā kupu rite.

$$2x^{2}+4x+2+2xx=5x\left(x+1\right)$$

Whakareatia te x ki te x, ka x^{2}.

$$2x^{2}+4x+2+2x^{2}=5x\left(x+1\right)$$

Pahekotia te 2x^{2} me 2x^{2}, ka 4x^{2}.

$$4x^{2}+4x+2=5x\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 5x ki te x+1.

$$4x^{2}+4x+2=5x^{2}+5x$$

Tangohia te 5x^{2} mai i ngā taha e rua.

$$4x^{2}+4x+2-5x^{2}=5x$$

Pahekotia te 4x^{2} me -5x^{2}, ka -x^{2}.

$$-x^{2}+4x+2=5x$$

Tangohia te 5x mai i ngā taha e rua.

$$-x^{2}+4x+2-5x=0$$

Pahekotia te 4x me -5x, ka -x.

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

Tangohia te 2 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

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

Whakawehea ngā taha e rua ki te -1.

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

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

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

Whakawehe -1 ki te -1.

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

Whakawehe -2 ki te -1.

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

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.

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

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

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

Tāpiri 2 ki te \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{9}{4}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{9}{4}$$

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

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Whakarūnātia.

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

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

$$x=1 x=-2$$