Whakaoti i te 2/x+15/x+6=1

$$\frac { 2 } { x } + \frac { 15 } { x + 6 } = 1$$

Whakaoti mō x

$$x=-1$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+6.

$$\left(x+6\right)\times 2+x\times 15=x\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+6 ki te 2.

$$2x+12+x\times 15=x\left(x+6\right)$$

Pahekotia te 2x me x\times 15, ka 17x.

$$17x+12=x\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+6.

$$17x+12=x^{2}+6x$$

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

$$17x+12-x^{2}=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$17x+12-x^{2}-6x=0$$

Pahekotia te 17x me -6x, ka 11x.

$$11x+12-x^{2}=0$$

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

$$-x^{2}+11x+12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=-12=-12$$

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 -12.

$$-1,12 -2,6 -3,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+12=11 -2+6=4 -3+4=1$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=12 b=-1$$

Tuhia anō te -x^{2}+11x+12 hei \left(-x^{2}+12x\right)+\left(-x+12\right).

$$\left(-x^{2}+12x\right)+\left(-x+12\right)$$

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

$$-x\left(x-12\right)-\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(-x-1\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te -x-1=0.

$$x=12 x=-1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+6.

$$\left(x+6\right)\times 2+x\times 15=x\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+6 ki te 2.

$$2x+12+x\times 15=x\left(x+6\right)$$

Pahekotia te 2x me x\times 15, ka 17x.

$$17x+12=x\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+6.

$$17x+12=x^{2}+6x$$

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

$$17x+12-x^{2}=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$17x+12-x^{2}-6x=0$$

Pahekotia te 17x me -6x, ka 11x.

$$11x+12-x^{2}=0$$

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.

$$-x^{2}+11x+12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 11 mō b, me 12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-11±\sqrt{121+4\times 12}}{2\left(-1\right)}$$

Whakareatia 4 ki te 12.

$$x=\frac{-11±\sqrt{121+48}}{2\left(-1\right)}$$

Tāpiri 121 ki te 48.

$$x=\frac{-11±\sqrt{169}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-11±13}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-11±13}{-2}$$

Nā, me whakaoti te whārite x=\frac{-11±13}{-2} ina he tāpiri te ±. Tāpiri -11 ki te 13.

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

Whakawehe 2 ki te -2.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-11±13}{-2} ina he tango te ±. Tango 13 mai i -11.

$$x=-\frac{24}{-2}$$

Whakawehe -24 ki te -2.

$$x=12$$

Kua oti te whārite te whakatau.

$$x=-1 x=12$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -6,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+6.

$$\left(x+6\right)\times 2+x\times 15=x\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+6 ki te 2.

$$2x+12+x\times 15=x\left(x+6\right)$$

Pahekotia te 2x me x\times 15, ka 17x.

$$17x+12=x\left(x+6\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+6.

$$17x+12=x^{2}+6x$$

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

$$17x+12-x^{2}=6x$$

Tangohia te 6x mai i ngā taha e rua.

$$17x+12-x^{2}-6x=0$$

Pahekotia te 17x me -6x, ka 11x.

$$11x+12-x^{2}=0$$

Tangohia te 12 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$11x-x^{2}=-12$$

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.

$$-x^{2}+11x=-12$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+11x}{-1}=-\frac{12}{-1}$$

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

$$x^{2}+\frac{11}{-1}x=-\frac{12}{-1}$$

Whakawehe 11 ki te -1.

$$x^{2}-11x=-\frac{12}{-1}$$

Whakawehe -12 ki te -1.

$$x^{2}-11x=12$$

Whakawehea te -11, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{2}. Nā, tāpiria te pūrua o te -\frac{11}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-11x+\left(-\frac{11}{2}\right)^{2}=12+\left(-\frac{11}{2}\right)^{2}$$

Pūruatia -\frac{11}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-11x+\frac{121}{4}=12+\frac{121}{4}$$

Tāpiri 12 ki te \frac{121}{4}.

$$x^{2}-11x+\frac{121}{4}=\frac{169}{4}$$

Tauwehea x^{2}-11x+\frac{121}{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}.

$$\left(x-\frac{11}{2}\right)^{2}=\frac{169}{4}$$

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

$$\sqrt{\left(x-\frac{11}{2}\right)^{2}}=\sqrt{\frac{169}{4}}$$

Whakarūnātia.

$$x-\frac{11}{2}=\frac{13}{2} x-\frac{11}{2}=-\frac{13}{2}$$

Me tāpiri \frac{11}{2} ki ngā taha e rua o te whārite.

$$x=12 x=-1$$