Whakaoti i te x^2+5x/x+1+x+1/1=10

$$\frac { x ^ { 2 } + 5 x } { x + 1 } + \frac { x + 1 } { 1 } = 10$$

Whakaoti mō x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+1.

$$x^{2}+5x+\left(x+1\right)\left(x+1\right)=10\left(x+1\right)$$

Whakareatia te x+1 ki te x+1, ka \left(x+1\right)^{2}.

$$x^{2}+5x+\left(x+1\right)^{2}=10\left(x+1\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$x^{2}+5x+x^{2}+2x+1=10\left(x+1\right)$$

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

$$2x^{2}+5x+2x+1=10\left(x+1\right)$$

Pahekotia te 5x me 2x, ka 7x.

$$2x^{2}+7x+1=10\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+1.

$$2x^{2}+7x+1=10x+10$$

Tangohia te 10x mai i ngā taha e rua.

$$2x^{2}+7x+1-10x=10$$

Pahekotia te 7x me -10x, ka -3x.

$$2x^{2}-3x+1=10$$

Tangohia te 10 mai i ngā taha e rua.

$$2x^{2}-3x+1-10=0$$

Tangohia te 10 i te 1, ka -9.

$$2x^{2}-3x-9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 2x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=2\left(-9\right)=-18$$

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 -18.

$$1,-18 2,-9 3,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-18=-17 2-9=-7 3-6=-3$$

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

$$a=-6 b=3$$

Tuhia anō te 2x^{2}-3x-9 hei \left(2x^{2}-6x\right)+\left(3x-9\right).

$$\left(2x^{2}-6x\right)+\left(3x-9\right)$$

Tauwehea te 2x i te tuatahi me te 3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+1.

$$x^{2}+5x+\left(x+1\right)\left(x+1\right)=10\left(x+1\right)$$

Whakareatia te x+1 ki te x+1, ka \left(x+1\right)^{2}.

$$x^{2}+5x+\left(x+1\right)^{2}=10\left(x+1\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$x^{2}+5x+x^{2}+2x+1=10\left(x+1\right)$$

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

$$2x^{2}+5x+2x+1=10\left(x+1\right)$$

Pahekotia te 5x me 2x, ka 7x.

$$2x^{2}+7x+1=10\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+1.

$$2x^{2}+7x+1=10x+10$$

Tangohia te 10x mai i ngā taha e rua.

$$2x^{2}+7x+1-10x=10$$

Pahekotia te 7x me -10x, ka -3x.

$$2x^{2}-3x+1=10$$

Tangohia te 10 mai i ngā taha e rua.

$$2x^{2}-3x+1-10=0$$

Tangohia te 10 i te 1, ka -9.

$$2x^{2}-3x-9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -3 mō b, me -9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

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

Pūrua -3.

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

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-3\right)±\sqrt{9-8\left(-9\right)}}{2\times 2}$$

Whakareatia -8 ki te -9.

$$x=\frac{-\left(-3\right)±\sqrt{9+72}}{2\times 2}$$

Tāpiri 9 ki te 72.

$$x=\frac{-\left(-3\right)±\sqrt{81}}{2\times 2}$$

Tuhia te pūtakerua o te 81.

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

Ko te tauaro o -3 ko 3.

$$x=\frac{3±9}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{3±9}{4}$$

Nā, me whakaoti te whārite x=\frac{3±9}{4} ina he tāpiri te ±. Tāpiri 3 ki te 9.

$$x=\frac{12}{4}$$

Whakawehe 12 ki te 4.

$$x=3$$

Nā, me whakaoti te whārite x=\frac{3±9}{4} ina he tango te ±. Tango 9 mai i 3.

$$x=-\frac{6}{4}$$

Whakahekea te hautanga \frac{-6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki -1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x+1.

$$x^{2}+5x+\left(x+1\right)\left(x+1\right)=10\left(x+1\right)$$

Whakareatia te x+1 ki te x+1, ka \left(x+1\right)^{2}.

$$x^{2}+5x+\left(x+1\right)^{2}=10\left(x+1\right)$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(x+1\right)^{2}.

$$x^{2}+5x+x^{2}+2x+1=10\left(x+1\right)$$

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

$$2x^{2}+5x+2x+1=10\left(x+1\right)$$

Pahekotia te 5x me 2x, ka 7x.

$$2x^{2}+7x+1=10\left(x+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10 ki te x+1.

$$2x^{2}+7x+1=10x+10$$

Tangohia te 10x mai i ngā taha e rua.

$$2x^{2}+7x+1-10x=10$$

Pahekotia te 7x me -10x, ka -3x.

$$2x^{2}-3x+1=10$$

Tangohia te 1 mai i ngā taha e rua.

$$2x^{2}-3x=10-1$$

Tangohia te 1 i te 10, ka 9.

$$2x^{2}-3x=9$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-3x}{2}=\frac{9}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}-\frac{3}{2}x=\frac{9}{2}$$

Whakawehea te -\frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{4}. Nā, tāpiria te pūrua o te -\frac{3}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{3}{2}x+\left(-\frac{3}{4}\right)^{2}=\frac{9}{2}+\left(-\frac{3}{4}\right)^{2}$$

Pūruatia -\frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{9}{2}+\frac{9}{16}$$

Tāpiri \frac{9}{2} ki te \frac{9}{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.

$$x^{2}-\frac{3}{2}x+\frac{9}{16}=\frac{81}{16}$$

Tauwehea x^{2}-\frac{3}{2}x+\frac{9}{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}.

$$\left(x-\frac{3}{4}\right)^{2}=\frac{81}{16}$$

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

$$\sqrt{\left(x-\frac{3}{4}\right)^{2}}=\sqrt{\frac{81}{16}}$$

Whakarūnātia.

$$x-\frac{3}{4}=\frac{9}{4} x-\frac{3}{4}=-\frac{9}{4}$$

Me tāpiri \frac{3}{4} ki ngā taha e rua o te whārite.

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