Whakaoti i te 168/x-168/x+14=1

$$\frac { 168 } { x } - \frac { 168 } { x + 14 } = 1$$

Whakaoti mō x

$$x=-56$$

$$x=42$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -14,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+14\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+14.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+14 ki te 168.

$$168x+2352-x\times 168=x\left(x+14\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+14.

$$168x+2352-x\times 168=x^{2}+14x$$

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

$$168x+2352-x\times 168-x^{2}=14x$$

Tangohia te 14x mai i ngā taha e rua.

$$168x+2352-x\times 168-x^{2}-14x=0$$

Pahekotia te 168x me -14x, ka 154x.

$$154x+2352-x\times 168-x^{2}=0$$

Whakareatia te -1 ki te 168, ka -168.

$$154x+2352-168x-x^{2}=0$$

Pahekotia te 154x me -168x, ka -14x.

$$-14x+2352-x^{2}=0$$

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

$$-x^{2}-14x+2352=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+2352. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=-2352=-2352$$

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 -2352.

$$1,-2352 2,-1176 3,-784 4,-588 6,-392 7,-336 8,-294 12,-196 14,-168 16,-147 21,-112 24,-98 28,-84 42,-56 48,-49$$

Tātaihia te tapeke mō ia takirua.

$$1-2352=-2351 2-1176=-1174 3-784=-781 4-588=-584 6-392=-386 7-336=-329 8-294=-286 12-196=-184 14-168=-154 16-147=-131 21-112=-91 24-98=-74 28-84=-56 42-56=-14 48-49=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=42 b=-56$$

Tuhia anō te -x^{2}-14x+2352 hei \left(-x^{2}+42x\right)+\left(-56x+2352\right).

$$\left(-x^{2}+42x\right)+\left(-56x+2352\right)$$

Tauwehea te x i te tuatahi me te 56 i te rōpū tuarua.

$$x\left(-x+42\right)+56\left(-x+42\right)$$

Whakatauwehea atu te kīanga pātahi -x+42 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+42\right)\left(x+56\right)$$

Hei kimi otinga whārite, me whakaoti te -x+42=0 me te x+56=0.

$$x=42 x=-56$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -14,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+14\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+14.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+14 ki te 168.

$$168x+2352-x\times 168=x\left(x+14\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+14.

$$168x+2352-x\times 168=x^{2}+14x$$

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

$$168x+2352-x\times 168-x^{2}=14x$$

Tangohia te 14x mai i ngā taha e rua.

$$168x+2352-x\times 168-x^{2}-14x=0$$

Pahekotia te 168x me -14x, ka 154x.

$$154x+2352-x\times 168-x^{2}=0$$

Whakareatia te -1 ki te 168, ka -168.

$$154x+2352-168x-x^{2}=0$$

Pahekotia te 154x me -168x, ka -14x.

$$-14x+2352-x^{2}=0$$

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.

$$-x^{2}-14x+2352=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -14 mō b, me 2352 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\left(-1\right)\times 2352}}{2\left(-1\right)}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\left(-1\right)\times 2352}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-14\right)±\sqrt{196+4\times 2352}}{2\left(-1\right)}$$

Whakareatia 4 ki te 2352.

$$x=\frac{-\left(-14\right)±\sqrt{196+9408}}{2\left(-1\right)}$$

Tāpiri 196 ki te 9408.

$$x=\frac{-\left(-14\right)±\sqrt{9604}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 9604.

$$x=\frac{-\left(-14\right)±98}{2\left(-1\right)}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±98}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{14±98}{-2}$$

Nā, me whakaoti te whārite x=\frac{14±98}{-2} ina he tāpiri te ±. Tāpiri 14 ki te 98.

$$x=\frac{112}{-2}$$

Whakawehe 112 ki te -2.

$$x=-56$$

Nā, me whakaoti te whārite x=\frac{14±98}{-2} ina he tango te ±. Tango 98 mai i 14.

$$x=-\frac{84}{-2}$$

Whakawehe -84 ki te -2.

$$x=42$$

Kua oti te whārite te whakatau.

$$x=-56 x=42$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -14,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+14\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+14.

$$\left(x+14\right)\times 168-x\times 168=x\left(x+14\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+14 ki te 168.

$$168x+2352-x\times 168=x\left(x+14\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+14.

$$168x+2352-x\times 168=x^{2}+14x$$

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

$$168x+2352-x\times 168-x^{2}=14x$$

Tangohia te 14x mai i ngā taha e rua.

$$168x+2352-x\times 168-x^{2}-14x=0$$

Pahekotia te 168x me -14x, ka 154x.

$$154x+2352-x\times 168-x^{2}=0$$

Tangohia te 2352 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$154x-x\times 168-x^{2}=-2352$$

Whakareatia te -1 ki te 168, ka -168.

$$154x-168x-x^{2}=-2352$$

Pahekotia te 154x me -168x, ka -14x.

$$-14x-x^{2}=-2352$$

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.

$$-x^{2}-14x=-2352$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-14x}{-1}=-\frac{2352}{-1}$$

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

$$x^{2}+\left(-\frac{14}{-1}\right)x=-\frac{2352}{-1}$$

Whakawehe -14 ki te -1.

$$x^{2}+14x=-\frac{2352}{-1}$$

Whakawehe -2352 ki te -1.

$$x^{2}+14x=2352$$

Whakawehea te 14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 7. Nā, tāpiria te pūrua o te 7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+14x+7^{2}=2352+7^{2}$$

Pūrua 7.

$$x^{2}+14x+49=2352+49$$

Tāpiri 2352 ki te 49.

$$x^{2}+14x+49=2401$$

Tauwehea x^{2}+14x+49. 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}.

$$\left(x+7\right)^{2}=2401$$

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

$$\sqrt{\left(x+7\right)^{2}}=\sqrt{2401}$$

Whakarūnātia.

$$x+7=49 x+7=-49$$

Me tango 7 mai i ngā taha e rua o te whārite.

$$x=42 x=-56$$