Whakaoti i te 360/x-360/x+5=1

$$\frac { 360 } { x } - \frac { 360 } { x + 5 } = 1$$

Whakaoti mō x

$$x=-45$$

$$x=40$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+5.

$$\left(x+5\right)\times 360-x\times 360=x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+5 ki te 360.

$$360x+1800-x\times 360=x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+5.

$$360x+1800-x\times 360=x^{2}+5x$$

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

$$360x+1800-x\times 360-x^{2}=5x$$

Tangohia te 5x mai i ngā taha e rua.

$$360x+1800-x\times 360-x^{2}-5x=0$$

Pahekotia te 360x me -5x, ka 355x.

$$355x+1800-x\times 360-x^{2}=0$$

Whakareatia te -1 ki te 360, ka -360.

$$355x+1800-360x-x^{2}=0$$

Pahekotia te 355x me -360x, ka -5x.

$$-5x+1800-x^{2}=0$$

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

$$-x^{2}-5x+1800=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+1800. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=-1800=-1800$$

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 -1800.

$$1,-1800 2,-900 3,-600 4,-450 5,-360 6,-300 8,-225 9,-200 10,-180 12,-150 15,-120 18,-100 20,-90 24,-75 25,-72 30,-60 36,-50 40,-45$$

Tātaihia te tapeke mō ia takirua.

$$1-1800=-1799 2-900=-898 3-600=-597 4-450=-446 5-360=-355 6-300=-294 8-225=-217 9-200=-191 10-180=-170 12-150=-138 15-120=-105 18-100=-82 20-90=-70 24-75=-51 25-72=-47 30-60=-30 36-50=-14 40-45=-5$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=40 b=-45$$

Tuhia anō te -x^{2}-5x+1800 hei \left(-x^{2}+40x\right)+\left(-45x+1800\right).

$$\left(-x^{2}+40x\right)+\left(-45x+1800\right)$$

Tauwehea te x i te tuatahi me te 45 i te rōpū tuarua.

$$x\left(-x+40\right)+45\left(-x+40\right)$$

Whakatauwehea atu te kīanga pātahi -x+40 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-x+40\right)\left(x+45\right)$$

Hei kimi otinga whārite, me whakaoti te -x+40=0 me te x+45=0.

$$x=40 x=-45$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+5.

$$\left(x+5\right)\times 360-x\times 360=x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+5 ki te 360.

$$360x+1800-x\times 360=x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+5.

$$360x+1800-x\times 360=x^{2}+5x$$

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

$$360x+1800-x\times 360-x^{2}=5x$$

Tangohia te 5x mai i ngā taha e rua.

$$360x+1800-x\times 360-x^{2}-5x=0$$

Pahekotia te 360x me -5x, ka 355x.

$$355x+1800-x\times 360-x^{2}=0$$

Whakareatia te -1 ki te 360, ka -360.

$$355x+1800-360x-x^{2}=0$$

Pahekotia te 355x me -360x, ka -5x.

$$-5x+1800-x^{2}=0$$

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.

$$-x^{2}-5x+1800=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -5 mō b, me 1800 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\left(-1\right)\times 1800}}{2\left(-1\right)}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\left(-1\right)\times 1800}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+4\times 1800}}{2\left(-1\right)}$$

Whakareatia 4 ki te 1800.

$$x=\frac{-\left(-5\right)±\sqrt{25+7200}}{2\left(-1\right)}$$

Tāpiri 25 ki te 7200.

$$x=\frac{-\left(-5\right)±\sqrt{7225}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 7225.

$$x=\frac{-\left(-5\right)±85}{2\left(-1\right)}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±85}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{5±85}{-2}$$

Nā, me whakaoti te whārite x=\frac{5±85}{-2} ina he tāpiri te ±. Tāpiri 5 ki te 85.

$$x=\frac{90}{-2}$$

Whakawehe 90 ki te -2.

$$x=-45$$

Nā, me whakaoti te whārite x=\frac{5±85}{-2} ina he tango te ±. Tango 85 mai i 5.

$$x=-\frac{80}{-2}$$

Whakawehe -80 ki te -2.

$$x=40$$

Kua oti te whārite te whakatau.

$$x=-45 x=40$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -5,0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te x\left(x+5\right), arā, te tauraro pātahi he tino iti rawa te kitea o x,x+5.

$$\left(x+5\right)\times 360-x\times 360=x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x+5 ki te 360.

$$360x+1800-x\times 360=x\left(x+5\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x ki te x+5.

$$360x+1800-x\times 360=x^{2}+5x$$

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

$$360x+1800-x\times 360-x^{2}=5x$$

Tangohia te 5x mai i ngā taha e rua.

$$360x+1800-x\times 360-x^{2}-5x=0$$

Pahekotia te 360x me -5x, ka 355x.

$$355x+1800-x\times 360-x^{2}=0$$

Tangohia te 1800 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$355x-x\times 360-x^{2}=-1800$$

Whakareatia te -1 ki te 360, ka -360.

$$355x-360x-x^{2}=-1800$$

Pahekotia te 355x me -360x, ka -5x.

$$-5x-x^{2}=-1800$$

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.

$$-x^{2}-5x=-1800$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-5x}{-1}=-\frac{1800}{-1}$$

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

$$x^{2}+\left(-\frac{5}{-1}\right)x=-\frac{1800}{-1}$$

Whakawehe -5 ki te -1.

$$x^{2}+5x=-\frac{1800}{-1}$$

Whakawehe -1800 ki te -1.

$$x^{2}+5x=1800$$

Whakawehea te 5, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{5}{2}. Nā, tāpiria te pūrua o te \frac{5}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+5x+\left(\frac{5}{2}\right)^{2}=1800+\left(\frac{5}{2}\right)^{2}$$

Pūruatia \frac{5}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+5x+\frac{25}{4}=1800+\frac{25}{4}$$

Tāpiri 1800 ki te \frac{25}{4}.

$$x^{2}+5x+\frac{25}{4}=\frac{7225}{4}$$

Tauwehea x^{2}+5x+\frac{25}{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}.

$$\left(x+\frac{5}{2}\right)^{2}=\frac{7225}{4}$$

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

$$\sqrt{\left(x+\frac{5}{2}\right)^{2}}=\sqrt{\frac{7225}{4}}$$

Whakarūnātia.

$$x+\frac{5}{2}=\frac{85}{2} x+\frac{5}{2}=-\frac{85}{2}$$

Me tango \frac{5}{2} mai i ngā taha e rua o te whārite.

$$x=40 x=-45$$