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

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

Whakaoti mō x

$$x=-\frac{1}{2}=-0.5$$

$$x=1$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

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

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

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

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

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

Pahekotia te 5x^{2} me x^{2}, ka 6x^{2}.

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

Pahekotia te x me x, ka 2x.

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

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

$$6x^{2}+2x-2=2x^{2}+4x$$

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

$$6x^{2}+2x-2-2x^{2}=4x$$

Pahekotia te 6x^{2} me -2x^{2}, ka 4x^{2}.

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

Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te 2x me -4x, ka -2x.

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

Whakawehea ngā taha e rua ki te 2.

$$2x^{2}-x-1=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2x^{2}+ax+bx-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=2\left(-1\right)=-2$$

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.

$$a=-2 b=1$$

Tuhia anō te 2x^{2}-x-1 hei \left(2x^{2}-2x\right)+\left(x-1\right).

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

Whakatauwehea atu 2x i te 2x^{2}-2x.

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

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

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

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

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

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

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

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

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

Pahekotia te 5x^{2} me x^{2}, ka 6x^{2}.

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

Pahekotia te x me x, ka 2x.

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

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

$$6x^{2}+2x-2=2x^{2}+4x$$

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

$$6x^{2}+2x-2-2x^{2}=4x$$

Pahekotia te 6x^{2} me -2x^{2}, ka 4x^{2}.

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

Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te 2x me -4x, ka -2x.

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

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

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

Pūrua -2.

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

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-2\right)±\sqrt{4-16\left(-2\right)}}{2\times 4}$$

Whakareatia -16 ki te -2.

$$x=\frac{-\left(-2\right)±\sqrt{4+32}}{2\times 4}$$

Tāpiri 4 ki te 32.

$$x=\frac{-\left(-2\right)±\sqrt{36}}{2\times 4}$$

Tuhia te pūtakerua o te 36.

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

Ko te tauaro o -2 ko 2.

$$x=\frac{2±6}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{2±6}{8}$$

Nā, me whakaoti te whārite x=\frac{2±6}{8} ina he tāpiri te ±. Tāpiri 2 ki te 6.

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

Whakawehe 8 ki te 8.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{2±6}{8} ina he tango te ±. Tango 6 mai i 2.

$$x=-\frac{4}{8}$$

Whakahekea te hautanga \frac{-4}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

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

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

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

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

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

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

Pahekotia te 5x^{2} me x^{2}, ka 6x^{2}.

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

Pahekotia te x me x, ka 2x.

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

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

$$6x^{2}+2x-2=2x^{2}+4x$$

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

$$6x^{2}+2x-2-2x^{2}=4x$$

Pahekotia te 6x^{2} me -2x^{2}, ka 4x^{2}.

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

Tangohia te 4x mai i ngā taha e rua.

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

Pahekotia te 2x me -4x, ka -2x.

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

Me tāpiri te 2 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

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

Whakawehea ngā taha e rua ki te 4.

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

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

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

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

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

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

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

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{2}+\frac{1}{16}$$

Tāpiri \frac{1}{2} ki te \frac{1}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{9}{16}$$

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

$$\left(x-\frac{1}{4}\right)^{2}=\frac{9}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{9}{16}}$$

Whakarūnātia.

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

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

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