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

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

Whakaoti mō x

$$x=\frac{5}{6}\approx 0.833333333$$

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 x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x^{2}+x.

$$x\left(x+1\right)+x\times 5x=5$$

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

$$x^{2}+x+x\times 5x=5$$

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

$$x^{2}+x+x^{2}\times 5=5$$

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

$$6x^{2}+x=5$$

Tangohia te 5 mai i ngā taha e rua.

$$6x^{2}+x-5=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 6x^{2}+ax+bx-5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=6\left(-5\right)=-30$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

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

$$a=-5 b=6$$

Tuhia anō te 6x^{2}+x-5 hei \left(6x^{2}-5x\right)+\left(6x-5\right).

$$\left(6x^{2}-5x\right)+\left(6x-5\right)$$

Whakatauwehea atu x i te 6x^{2}-5x.

$$x\left(6x-5\right)+6x-5$$

Whakatauwehea atu te kīanga pātahi 6x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(6x-5\right)\left(x+1\right)$$

Hei kimi otinga whārite, me whakaoti te 6x-5=0 me te x+1=0.

$$x=\frac{5}{6} x=-1$$

Tē taea kia ōrite te tāupe x ki -1.

$$x=\frac{5}{6}$$

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 x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x^{2}+x.

$$x\left(x+1\right)+x\times 5x=5$$

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

$$x^{2}+x+x\times 5x=5$$

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

$$x^{2}+x+x^{2}\times 5=5$$

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

$$6x^{2}+x=5$$

Tangohia te 5 mai i ngā taha e rua.

$$6x^{2}+x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 1 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-1±\sqrt{1^{2}-4\times 6\left(-5\right)}}{2\times 6}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\times 6\left(-5\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-1±\sqrt{1-24\left(-5\right)}}{2\times 6}$$

Whakareatia -24 ki te -5.

$$x=\frac{-1±\sqrt{1+120}}{2\times 6}$$

Tāpiri 1 ki te 120.

$$x=\frac{-1±\sqrt{121}}{2\times 6}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-1±11}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-1±11}{12}$$

Nā, me whakaoti te whārite x=\frac{-1±11}{12} ina he tāpiri te ±. Tāpiri -1 ki te 11.

$$x=\frac{10}{12}$$

Whakahekea te hautanga \frac{10}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{5}{6}$$

Nā, me whakaoti te whārite x=\frac{-1±11}{12} ina he tango te ±. Tango 11 mai i -1.

$$x=-\frac{12}{12}$$

Whakawehe -12 ki te 12.

$$x=-1$$

Kua oti te whārite te whakatau.

$$x=\frac{5}{6} x=-1$$

Tē taea kia ōrite te tāupe x ki -1.

$$x=\frac{5}{6}$$

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 x\left(x+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x+1,x^{2}+x.

$$x\left(x+1\right)+x\times 5x=5$$

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

$$x^{2}+x+x\times 5x=5$$

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

$$x^{2}+x+x^{2}\times 5=5$$

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

$$6x^{2}+x=5$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+x}{6}=\frac{5}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{1}{6}x=\frac{5}{6}$$

Whakawehea te \frac{1}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{12}. Nā, tāpiria te pūrua o te \frac{1}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{6}x+\left(\frac{1}{12}\right)^{2}=\frac{5}{6}+\left(\frac{1}{12}\right)^{2}$$

Pūruatia \frac{1}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{6}x+\frac{1}{144}=\frac{5}{6}+\frac{1}{144}$$

Tāpiri \frac{5}{6} ki te \frac{1}{144} 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.

$$x^{2}+\frac{1}{6}x+\frac{1}{144}=\frac{121}{144}$$

Tauwehea x^{2}+\frac{1}{6}x+\frac{1}{144}. 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}.

$$\left(x+\frac{1}{12}\right)^{2}=\frac{121}{144}$$

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

$$\sqrt{\left(x+\frac{1}{12}\right)^{2}}=\sqrt{\frac{121}{144}}$$

Whakarūnātia.

$$x+\frac{1}{12}=\frac{11}{12} x+\frac{1}{12}=-\frac{11}{12}$$

Me tango \frac{1}{12} mai i ngā taha e rua o te whārite.

$$x=\frac{5}{6} x=-1$$

Tē taea kia ōrite te tāupe x ki -1.

$$x=\frac{5}{6}$$