Whakaoti i te 560/x-2-450/x=10

$$\frac { 560 } { x - 2 } - \frac { 450 } { x } = 10$$

Whakaoti mō x

$$x=-5$$

$$x=18$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,2 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-2,x.

$$x\times 560-\left(x-2\right)\times 450=10x\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 450.

$$x\times 560-\left(450x-900\right)=10x\left(x-2\right)$$

Hei kimi i te tauaro o 450x-900, kimihia te tauaro o ia taurangi.

$$x\times 560-450x+900=10x\left(x-2\right)$$

Pahekotia te x\times 560 me -450x, ka 110x.

$$110x+900=10x\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x ki te x-2.

$$110x+900=10x^{2}-20x$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$110x+900-10x^{2}=-20x$$

Me tāpiri te 20x ki ngā taha e rua.

$$110x+900-10x^{2}+20x=0$$

Pahekotia te 110x me 20x, ka 130x.

$$130x+900-10x^{2}=0$$

Whakawehea ngā taha e rua ki te 10.

$$13x+90-x^{2}=0$$

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

$$-x^{2}+13x+90=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+90. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=13 ab=-90=-90$$

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 -90.

$$-1,90 -2,45 -3,30 -5,18 -6,15 -9,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+90=89 -2+45=43 -3+30=27 -5+18=13 -6+15=9 -9+10=1$$

Ko te otinga te takirua ka hoatu i te tapeke 13.

$$a=18 b=-5$$

Tuhia anō te -x^{2}+13x+90 hei \left(-x^{2}+18x\right)+\left(-5x+90\right).

$$\left(-x^{2}+18x\right)+\left(-5x+90\right)$$

Tauwehea te -x i te tuatahi me te -5 i te rōpū tuarua.

$$-x\left(x-18\right)-5\left(x-18\right)$$

Whakatauwehea atu te kīanga pātahi x-18 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-18\right)\left(-x-5\right)$$

Hei kimi otinga whārite, me whakaoti te x-18=0 me te -x-5=0.

$$x=18 x=-5$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,2 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-2,x.

$$x\times 560-\left(x-2\right)\times 450=10x\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 450.

$$x\times 560-\left(450x-900\right)=10x\left(x-2\right)$$

Hei kimi i te tauaro o 450x-900, kimihia te tauaro o ia taurangi.

$$x\times 560-450x+900=10x\left(x-2\right)$$

Pahekotia te x\times 560 me -450x, ka 110x.

$$110x+900=10x\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x ki te x-2.

$$110x+900=10x^{2}-20x$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$110x+900-10x^{2}=-20x$$

Me tāpiri te 20x ki ngā taha e rua.

$$110x+900-10x^{2}+20x=0$$

Pahekotia te 110x me 20x, ka 130x.

$$130x+900-10x^{2}=0$$

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.

$$-10x^{2}+130x+900=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -10 mō a, 130 mō b, me 900 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-130±\sqrt{130^{2}-4\left(-10\right)\times 900}}{2\left(-10\right)}$$

Pūrua 130.

$$x=\frac{-130±\sqrt{16900-4\left(-10\right)\times 900}}{2\left(-10\right)}$$

Whakareatia -4 ki te -10.

$$x=\frac{-130±\sqrt{16900+40\times 900}}{2\left(-10\right)}$$

Whakareatia 40 ki te 900.

$$x=\frac{-130±\sqrt{16900+36000}}{2\left(-10\right)}$$

Tāpiri 16900 ki te 36000.

$$x=\frac{-130±\sqrt{52900}}{2\left(-10\right)}$$

Tuhia te pūtakerua o te 52900.

$$x=\frac{-130±230}{2\left(-10\right)}$$

Whakareatia 2 ki te -10.

$$x=\frac{-130±230}{-20}$$

Nā, me whakaoti te whārite x=\frac{-130±230}{-20} ina he tāpiri te ±. Tāpiri -130 ki te 230.

$$x=\frac{100}{-20}$$

Whakawehe 100 ki te -20.

$$x=-5$$

Nā, me whakaoti te whārite x=\frac{-130±230}{-20} ina he tango te ±. Tango 230 mai i -130.

$$x=-\frac{360}{-20}$$

Whakawehe -360 ki te -20.

$$x=18$$

Kua oti te whārite te whakatau.

$$x=-5 x=18$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara 0,2 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-2,x.

$$x\times 560-\left(x-2\right)\times 450=10x\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te x-2 ki te 450.

$$x\times 560-\left(450x-900\right)=10x\left(x-2\right)$$

Hei kimi i te tauaro o 450x-900, kimihia te tauaro o ia taurangi.

$$x\times 560-450x+900=10x\left(x-2\right)$$

Pahekotia te x\times 560 me -450x, ka 110x.

$$110x+900=10x\left(x-2\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10x ki te x-2.

$$110x+900=10x^{2}-20x$$

Tangohia te 10x^{2} mai i ngā taha e rua.

$$110x+900-10x^{2}=-20x$$

Me tāpiri te 20x ki ngā taha e rua.

$$110x+900-10x^{2}+20x=0$$

Pahekotia te 110x me 20x, ka 130x.

$$130x+900-10x^{2}=0$$

Tangohia te 900 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$130x-10x^{2}=-900$$

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.

$$-10x^{2}+130x=-900$$

Whakawehea ngā taha e rua ki te -10.

$$\frac{-10x^{2}+130x}{-10}=-\frac{900}{-10}$$

Mā te whakawehe ki te -10 ka wetekia te whakareanga ki te -10.

$$x^{2}+\frac{130}{-10}x=-\frac{900}{-10}$$

Whakawehe 130 ki te -10.

$$x^{2}-13x=-\frac{900}{-10}$$

Whakawehe -900 ki te -10.

$$x^{2}-13x=90$$

Whakawehea te -13, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{13}{2}. Nā, tāpiria te pūrua o te -\frac{13}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-13x+\left(-\frac{13}{2}\right)^{2}=90+\left(-\frac{13}{2}\right)^{2}$$

Pūruatia -\frac{13}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-13x+\frac{169}{4}=90+\frac{169}{4}$$

Tāpiri 90 ki te \frac{169}{4}.

$$x^{2}-13x+\frac{169}{4}=\frac{529}{4}$$

Tauwehea x^{2}-13x+\frac{169}{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}.

$$\left(x-\frac{13}{2}\right)^{2}=\frac{529}{4}$$

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

$$\sqrt{\left(x-\frac{13}{2}\right)^{2}}=\sqrt{\frac{529}{4}}$$

Whakarūnātia.

$$x-\frac{13}{2}=\frac{23}{2} x-\frac{13}{2}=-\frac{23}{2}$$

Me tāpiri \frac{13}{2} ki ngā taha e rua o te whārite.

$$x=18 x=-5$$