Whakaoti i te 3/x+5/x+2=2

$$\frac { 3 } { x } + \frac { 5 } { x + 2 } = 2$$

Whakaoti mō x

$$x=-1$$

$$x=3$$

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,x+2.

$$\left(x+2\right)\times 3+x\times 5=2x\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 3.

$$3x+6+x\times 5=2x\left(x+2\right)$$

Pahekotia te 3x me x\times 5, ka 8x.

$$8x+6=2x\left(x+2\right)$$

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

$$8x+6=2x^{2}+4x$$

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

$$8x+6-2x^{2}=4x$$

Tangohia te 4x mai i ngā taha e rua.

$$8x+6-2x^{2}-4x=0$$

Pahekotia te 8x me -4x, ka 4x.

$$4x+6-2x^{2}=0$$

Whakawehea ngā taha e rua ki te 2.

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

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

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=-3=-3$$

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. Ko te takirua anake pērā ko te otinga pūnaha.

$$a=3 b=-1$$

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

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

Tauwehea te -x i te tuatahi me te -1 i te rōpū tuarua.

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

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

$$\left(x-3\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te -x-1=0.

$$x=3 x=-1$$

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,x+2.

$$\left(x+2\right)\times 3+x\times 5=2x\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 3.

$$3x+6+x\times 5=2x\left(x+2\right)$$

Pahekotia te 3x me x\times 5, ka 8x.

$$8x+6=2x\left(x+2\right)$$

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

$$8x+6=2x^{2}+4x$$

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

$$8x+6-2x^{2}=4x$$

Tangohia te 4x mai i ngā taha e rua.

$$8x+6-2x^{2}-4x=0$$

Pahekotia te 8x me -4x, ka 4x.

$$4x+6-2x^{2}=0$$

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.

$$-2x^{2}+4x+6=0$$

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

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

Pūrua 4.

$$x=\frac{-4±\sqrt{16-4\left(-2\right)\times 6}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-4±\sqrt{16+8\times 6}}{2\left(-2\right)}$$

Whakareatia 8 ki te 6.

$$x=\frac{-4±\sqrt{16+48}}{2\left(-2\right)}$$

Tāpiri 16 ki te 48.

$$x=\frac{-4±\sqrt{64}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 64.

$$x=\frac{-4±8}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{-4±8}{-4}$$

Nā, me whakaoti te whārite x=\frac{-4±8}{-4} ina he tāpiri te ±. Tāpiri -4 ki te 8.

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

Whakawehe 4 ki te -4.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-4±8}{-4} ina he tango te ±. Tango 8 mai i -4.

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

Whakawehe -12 ki te -4.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=-1 x=3$$

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,x+2.

$$\left(x+2\right)\times 3+x\times 5=2x\left(x+2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+2 ki te 3.

$$3x+6+x\times 5=2x\left(x+2\right)$$

Pahekotia te 3x me x\times 5, ka 8x.

$$8x+6=2x\left(x+2\right)$$

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

$$8x+6=2x^{2}+4x$$

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

$$8x+6-2x^{2}=4x$$

Tangohia te 4x mai i ngā taha e rua.

$$8x+6-2x^{2}-4x=0$$

Pahekotia te 8x me -4x, ka 4x.

$$4x+6-2x^{2}=0$$

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

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

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.

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

Whakawehea ngā taha e rua ki te -2.

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

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

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

Whakawehe 4 ki te -2.

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

Whakawehe -6 ki te -2.

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

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

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

Tāpiri 3 ki te 1.

$$x^{2}-2x+1=4$$

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

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

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

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

Whakarūnātia.

$$x-1=2 x-1=-2$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=3 x=-1$$