Whakaoti i te p-1/p+3-2/p-3=7-3p/p^2-9

$$\frac { p - 1 } { p + 3 } - \frac { 2 } { p - 3 } = \frac { 7 - 3 p } { p ^ { 2 } - 9 }$$

Whakaoti mō p

$$p=-2$$

$$p=5$$

Ngā Upane Whakamahi Tauwehe

Tē taea kia ōrite te tāupe p ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(p-3\right)\left(p+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o p+3,p-3,p^{2}-9.

$$\left(p-3\right)\left(p-1\right)-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tuaritanga hei whakarea te p-3 ki te p-1 ka whakakotahi i ngā kupu rite.

$$p^{2}-4p+3-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tohatoha hei whakarea te p+3 ki te 2.

$$p^{2}-4p+3-\left(2p+6\right)=7-3p$$

Hei kimi i te tauaro o 2p+6, kimihia te tauaro o ia taurangi.

$$p^{2}-4p+3-2p-6=7-3p$$

Pahekotia te -4p me -2p, ka -6p.

$$p^{2}-6p+3-6=7-3p$$

Tangohia te 6 i te 3, ka -3.

$$p^{2}-6p-3=7-3p$$

Tangohia te 7 mai i ngā taha e rua.

$$p^{2}-6p-3-7=-3p$$

Tangohia te 7 i te -3, ka -10.

$$p^{2}-6p-10=-3p$$

Me tāpiri te 3p ki ngā taha e rua.

$$p^{2}-6p-10+3p=0$$

Pahekotia te -6p me 3p, ka -3p.

$$p^{2}-3p-10=0$$

Hei whakaoti i te whārite, whakatauwehea te p^{2}-3p-10 mā te whakamahi i te tātai p^{2}+\left(a+b\right)p+ab=\left(p+a\right)\left(p+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=-10$$

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 -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-5 b=2$$

Me tuhi anō te kīanga whakatauwehe \left(p+a\right)\left(p+b\right) mā ngā uara i tātaihia.

$$\left(p-5\right)\left(p+2\right)$$

Hei kimi otinga whārite, me whakaoti te p-5=0 me te p+2=0.

$$p=5 p=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tē taea kia ōrite te tāupe p ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(p-3\right)\left(p+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o p+3,p-3,p^{2}-9.

$$\left(p-3\right)\left(p-1\right)-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tuaritanga hei whakarea te p-3 ki te p-1 ka whakakotahi i ngā kupu rite.

$$p^{2}-4p+3-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tohatoha hei whakarea te p+3 ki te 2.

$$p^{2}-4p+3-\left(2p+6\right)=7-3p$$

Hei kimi i te tauaro o 2p+6, kimihia te tauaro o ia taurangi.

$$p^{2}-4p+3-2p-6=7-3p$$

Pahekotia te -4p me -2p, ka -6p.

$$p^{2}-6p+3-6=7-3p$$

Tangohia te 6 i te 3, ka -3.

$$p^{2}-6p-3=7-3p$$

Tangohia te 7 mai i ngā taha e rua.

$$p^{2}-6p-3-7=-3p$$

Tangohia te 7 i te -3, ka -10.

$$p^{2}-6p-10=-3p$$

Me tāpiri te 3p ki ngā taha e rua.

$$p^{2}-6p-10+3p=0$$

Pahekotia te -6p me 3p, ka -3p.

$$p^{2}-3p-10=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei p^{2}+ap+bp-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-10\right)=-10$$

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 -10.

$$1,-10 2,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-10=-9 2-5=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-5 b=2$$

Tuhia anō te p^{2}-3p-10 hei \left(p^{2}-5p\right)+\left(2p-10\right).

$$\left(p^{2}-5p\right)+\left(2p-10\right)$$

Tauwehea te p i te tuatahi me te 2 i te rōpū tuarua.

$$p\left(p-5\right)+2\left(p-5\right)$$

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

$$\left(p-5\right)\left(p+2\right)$$

Hei kimi otinga whārite, me whakaoti te p-5=0 me te p+2=0.

$$p=5 p=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe p ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(p-3\right)\left(p+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o p+3,p-3,p^{2}-9.

$$\left(p-3\right)\left(p-1\right)-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tuaritanga hei whakarea te p-3 ki te p-1 ka whakakotahi i ngā kupu rite.

$$p^{2}-4p+3-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tohatoha hei whakarea te p+3 ki te 2.

$$p^{2}-4p+3-\left(2p+6\right)=7-3p$$

Hei kimi i te tauaro o 2p+6, kimihia te tauaro o ia taurangi.

$$p^{2}-4p+3-2p-6=7-3p$$

Pahekotia te -4p me -2p, ka -6p.

$$p^{2}-6p+3-6=7-3p$$

Tangohia te 6 i te 3, ka -3.

$$p^{2}-6p-3=7-3p$$

Tangohia te 7 mai i ngā taha e rua.

$$p^{2}-6p-3-7=-3p$$

Tangohia te 7 i te -3, ka -10.

$$p^{2}-6p-10=-3p$$

Me tāpiri te 3p ki ngā taha e rua.

$$p^{2}-6p-10+3p=0$$

Pahekotia te -6p me 3p, ka -3p.

$$p^{2}-3p-10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -3 mō b, me -10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-10\right)}}{2}$$

Pūrua -3.

$$p=\frac{-\left(-3\right)±\sqrt{9-4\left(-10\right)}}{2}$$

Whakareatia -4 ki te -10.

$$p=\frac{-\left(-3\right)±\sqrt{9+40}}{2}$$

Tāpiri 9 ki te 40.

$$p=\frac{-\left(-3\right)±\sqrt{49}}{2}$$

Tuhia te pūtakerua o te 49.

$$p=\frac{-\left(-3\right)±7}{2}$$

Ko te tauaro o -3 ko 3.

$$p=\frac{3±7}{2}$$

Nā, me whakaoti te whārite p=\frac{3±7}{2} ina he tāpiri te ±. Tāpiri 3 ki te 7.

$$p=\frac{10}{2}$$

Whakawehe 10 ki te 2.

$$p=5$$

Nā, me whakaoti te whārite p=\frac{3±7}{2} ina he tango te ±. Tango 7 mai i 3.

$$p=-\frac{4}{2}$$

Whakawehe -4 ki te 2.

$$p=-2$$

Kua oti te whārite te whakatau.

$$p=5 p=-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe p ki tētahi o ngā uara -3,3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(p-3\right)\left(p+3\right), arā, te tauraro pātahi he tino iti rawa te kitea o p+3,p-3,p^{2}-9.

$$\left(p-3\right)\left(p-1\right)-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tuaritanga hei whakarea te p-3 ki te p-1 ka whakakotahi i ngā kupu rite.

$$p^{2}-4p+3-\left(p+3\right)\times 2=7-3p$$

Whakamahia te āhuatanga tohatoha hei whakarea te p+3 ki te 2.

$$p^{2}-4p+3-\left(2p+6\right)=7-3p$$

Hei kimi i te tauaro o 2p+6, kimihia te tauaro o ia taurangi.

$$p^{2}-4p+3-2p-6=7-3p$$

Pahekotia te -4p me -2p, ka -6p.

$$p^{2}-6p+3-6=7-3p$$

Tangohia te 6 i te 3, ka -3.

$$p^{2}-6p-3=7-3p$$

Me tāpiri te 3p ki ngā taha e rua.

$$p^{2}-6p-3+3p=7$$

Pahekotia te -6p me 3p, ka -3p.

$$p^{2}-3p-3=7$$

Me tāpiri te 3 ki ngā taha e rua.

$$p^{2}-3p=7+3$$

Tāpirihia te 7 ki te 3, ka 10.

$$p^{2}-3p=10$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$p^{2}-3p+\left(-\frac{3}{2}\right)^{2}=10+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$p^{2}-3p+\frac{9}{4}=10+\frac{9}{4}$$

Tāpiri 10 ki te \frac{9}{4}.

$$p^{2}-3p+\frac{9}{4}=\frac{49}{4}$$

Tauwehea p^{2}-3p+\frac{9}{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}.

$$\left(p-\frac{3}{2}\right)^{2}=\frac{49}{4}$$

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

$$\sqrt{\left(p-\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Whakarūnātia.

$$p-\frac{3}{2}=\frac{7}{2} p-\frac{3}{2}=-\frac{7}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$p=5 p=-2$$