Whakaoti i te 1/x+1/x+6-1/4=0

$$\frac { 1 } { x } + \frac { 1 } { x + 6 } - \frac { 1 } { 4 } = 0$$

Whakaoti mō x

$$x=-4$$

$$x=6$$

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 4x\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+6,4.

$$4x+24+4x+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

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

$$8x+24+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Whakareatia te 4 ki te -\frac{1}{4}, ka -1.

$$8x+24-x\left(x+6\right)=0$$

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

$$8x+24-x^{2}-6x=0$$

Pahekotia te 8x me -6x, ka 2x.

$$2x+24-x^{2}=0$$

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

$$-x^{2}+2x+24=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+24. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=2 ab=-24=-24$$

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 -24.

$$-1,24 -2,12 -3,8 -4,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=6 b=-4$$

Tuhia anō te -x^{2}+2x+24 hei \left(-x^{2}+6x\right)+\left(-4x+24\right).

$$\left(-x^{2}+6x\right)+\left(-4x+24\right)$$

Tauwehea te -x i te tuatahi me te -4 i te rōpū tuarua.

$$-x\left(x-6\right)-4\left(x-6\right)$$

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

$$\left(x-6\right)\left(-x-4\right)$$

Hei kimi otinga whārite, me whakaoti te x-6=0 me te -x-4=0.

$$x=6 x=-4$$

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 4x\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+6,4.

$$4x+24+4x+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

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

$$8x+24+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Whakareatia te 4 ki te -\frac{1}{4}, ka -1.

$$8x+24-x\left(x+6\right)=0$$

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

$$8x+24-x^{2}-6x=0$$

Pahekotia te 8x me -6x, ka 2x.

$$2x+24-x^{2}=0$$

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.

$$-x^{2}+2x+24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, 2 mō b, me 24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-1\right)\times 24}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-2±\sqrt{4+4\times 24}}{2\left(-1\right)}$$

Whakareatia 4 ki te 24.

$$x=\frac{-2±\sqrt{4+96}}{2\left(-1\right)}$$

Tāpiri 4 ki te 96.

$$x=\frac{-2±\sqrt{100}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-2±10}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-2±10}{-2}$$

Nā, me whakaoti te whārite x=\frac{-2±10}{-2} ina he tāpiri te ±. Tāpiri -2 ki te 10.

$$x=\frac{8}{-2}$$

Whakawehe 8 ki te -2.

$$x=-4$$

Nā, me whakaoti te whārite x=\frac{-2±10}{-2} ina he tango te ±. Tango 10 mai i -2.

$$x=-\frac{12}{-2}$$

Whakawehe -12 ki te -2.

$$x=6$$

Kua oti te whārite te whakatau.

$$x=-4 x=6$$

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 4x\left(x+6\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+6,4.

$$4x+24+4x+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

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

$$8x+24+4x\left(x+6\right)\left(-\frac{1}{4}\right)=0$$

Whakareatia te 4 ki te -\frac{1}{4}, ka -1.

$$8x+24-x\left(x+6\right)=0$$

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

$$8x+24-x^{2}-6x=0$$

Pahekotia te 8x me -6x, ka 2x.

$$2x+24-x^{2}=0$$

Tangohia te 24 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-x^{2}=-24$$

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.

$$-x^{2}+2x=-24$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+2x}{-1}=-\frac{24}{-1}$$

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

$$x^{2}+\frac{2}{-1}x=-\frac{24}{-1}$$

Whakawehe 2 ki te -1.

$$x^{2}-2x=-\frac{24}{-1}$$

Whakawehe -24 ki te -1.

$$x^{2}-2x=24$$

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.

$$x^{2}-2x+1=24+1$$

Tāpiri 24 ki te 1.

$$x^{2}-2x+1=25$$

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}.

$$\left(x-1\right)^{2}=25$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x-1=5 x-1=-5$$

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

$$x=6 x=-4$$