Whakaoti i te 5/x-3/2=x/5

$$\frac { 5 } { x } - \frac { 3 } { 2 } = \frac { x } { 5 }$$

Whakaoti mō x

$$x = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

$$x=-10$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 10x, arā, te tauraro pātahi he tino iti rawa te kitea o x,2,5.

$$10\times 5+10x\left(-\frac{3}{2}\right)=2xx$$

Whakareatia te 10 ki te 5, ka 50.

$$50+10x\left(-\frac{3}{2}\right)=2xx$$

Tuhia te 10\left(-\frac{3}{2}\right) hei hautanga kotahi.

$$50+\frac{10\left(-3\right)}{2}x=2xx$$

Whakareatia te 10 ki te -3, ka -30.

$$50+\frac{-30}{2}x=2xx$$

Whakawehea te -30 ki te 2, kia riro ko -15.

$$50-15x=2xx$$

Whakareatia te x ki te x, ka x^{2}.

$$50-15x=2x^{2}$$

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

$$50-15x-2x^{2}=0$$

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

$$-2x^{2}-15x+50=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -2x^{2}+ax+bx+50. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-15 ab=-2\times 50=-100$$

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 -100.

$$1,-100 2,-50 4,-25 5,-20 10,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-100=-99 2-50=-48 4-25=-21 5-20=-15 10-10=0$$

Ko te otinga te takirua ka hoatu i te tapeke -15.

$$a=5 b=-20$$

Tuhia anō te -2x^{2}-15x+50 hei \left(-2x^{2}+5x\right)+\left(-20x+50\right).

$$\left(-2x^{2}+5x\right)+\left(-20x+50\right)$$

Tauwehea te -x i te tuatahi me te -10 i te rōpū tuarua.

$$-x\left(2x-5\right)-10\left(2x-5\right)$$

Whakatauwehea atu te kīanga pātahi 2x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-5\right)\left(-x-10\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-5=0 me te -x-10=0.

$$x=\frac{5}{2} x=-10$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 10x, arā, te tauraro pātahi he tino iti rawa te kitea o x,2,5.

$$10\times 5+10x\left(-\frac{3}{2}\right)=2xx$$

Whakareatia te 10 ki te 5, ka 50.

$$50+10x\left(-\frac{3}{2}\right)=2xx$$

Tuhia te 10\left(-\frac{3}{2}\right) hei hautanga kotahi.

$$50+\frac{10\left(-3\right)}{2}x=2xx$$

Whakareatia te 10 ki te -3, ka -30.

$$50+\frac{-30}{2}x=2xx$$

Whakawehea te -30 ki te 2, kia riro ko -15.

$$50-15x=2xx$$

Whakareatia te x ki te x, ka x^{2}.

$$50-15x=2x^{2}$$

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

$$50-15x-2x^{2}=0$$

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.

$$-2x^{2}-15x+50=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -2 mō a, -15 mō b, me 50 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-2\right)\times 50}}{2\left(-2\right)}$$

Pūrua -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-2\right)\times 50}}{2\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$x=\frac{-\left(-15\right)±\sqrt{225+8\times 50}}{2\left(-2\right)}$$

Whakareatia 8 ki te 50.

$$x=\frac{-\left(-15\right)±\sqrt{225+400}}{2\left(-2\right)}$$

Tāpiri 225 ki te 400.

$$x=\frac{-\left(-15\right)±\sqrt{625}}{2\left(-2\right)}$$

Tuhia te pūtakerua o te 625.

$$x=\frac{-\left(-15\right)±25}{2\left(-2\right)}$$

Ko te tauaro o -15 ko 15.

$$x=\frac{15±25}{2\left(-2\right)}$$

Whakareatia 2 ki te -2.

$$x=\frac{15±25}{-4}$$

Nā, me whakaoti te whārite x=\frac{15±25}{-4} ina he tāpiri te ±. Tāpiri 15 ki te 25.

$$x=\frac{40}{-4}$$

Whakawehe 40 ki te -4.

$$x=-10$$

Nā, me whakaoti te whārite x=\frac{15±25}{-4} ina he tango te ±. Tango 25 mai i 15.

$$x=-\frac{10}{-4}$$

Whakahekea te hautanga \frac{-10}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{5}{2}$$

Kua oti te whārite te whakatau.

$$x=-10 x=\frac{5}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te 10x, arā, te tauraro pātahi he tino iti rawa te kitea o x,2,5.

$$10\times 5+10x\left(-\frac{3}{2}\right)=2xx$$

Whakareatia te 10 ki te 5, ka 50.

$$50+10x\left(-\frac{3}{2}\right)=2xx$$

Tuhia te 10\left(-\frac{3}{2}\right) hei hautanga kotahi.

$$50+\frac{10\left(-3\right)}{2}x=2xx$$

Whakareatia te 10 ki te -3, ka -30.

$$50+\frac{-30}{2}x=2xx$$

Whakawehea te -30 ki te 2, kia riro ko -15.

$$50-15x=2xx$$

Whakareatia te x ki te x, ka x^{2}.

$$50-15x=2x^{2}$$

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

$$50-15x-2x^{2}=0$$

Tangohia te 50 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-15x-2x^{2}=-50$$

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.

$$-2x^{2}-15x=-50$$

Whakawehea ngā taha e rua ki te -2.

$$\frac{-2x^{2}-15x}{-2}=-\frac{50}{-2}$$

Mā te whakawehe ki te -2 ka wetekia te whakareanga ki te -2.

$$x^{2}+\left(-\frac{15}{-2}\right)x=-\frac{50}{-2}$$

Whakawehe -15 ki te -2.

$$x^{2}+\frac{15}{2}x=-\frac{50}{-2}$$

Whakawehe -50 ki te -2.

$$x^{2}+\frac{15}{2}x=25$$

Whakawehea te \frac{15}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{4}. Nā, tāpiria te pūrua o te \frac{15}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{15}{2}x+\left(\frac{15}{4}\right)^{2}=25+\left(\frac{15}{4}\right)^{2}$$

Pūruatia \frac{15}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{15}{2}x+\frac{225}{16}=25+\frac{225}{16}$$

Tāpiri 25 ki te \frac{225}{16}.

$$x^{2}+\frac{15}{2}x+\frac{225}{16}=\frac{625}{16}$$

Tauwehea x^{2}+\frac{15}{2}x+\frac{225}{16}. 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}.

$$\left(x+\frac{15}{4}\right)^{2}=\frac{625}{16}$$

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

$$\sqrt{\left(x+\frac{15}{4}\right)^{2}}=\sqrt{\frac{625}{16}}$$

Whakarūnātia.

$$x+\frac{15}{4}=\frac{25}{4} x+\frac{15}{4}=-\frac{25}{4}$$

Me tango \frac{15}{4} mai i ngā taha e rua o te whārite.

$$x=\frac{5}{2} x=-10$$