Whakaoti i te p^2=4p+12

$$p ^ { 2 } = 4 p + 12$$

Whakaoti mō p

$$p=-2$$

$$p=6$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 4p mai i ngā taha e rua.

$$p^{2}-4p=12$$

Tangohia te 12 mai i ngā taha e rua.

$$p^{2}-4p-12=0$$

Hei whakaoti i te whārite, whakatauwehea te p^{2}-4p-12 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.

$$a+b=-4 ab=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-6 b=2$$

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

$$\left(p-6\right)\left(p+2\right)$$

Hei kimi otinga whārite, me whakaoti te p-6=0 me te p+2=0.

$$p=6 p=-2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 4p mai i ngā taha e rua.

$$p^{2}-4p=12$$

Tangohia te 12 mai i ngā taha e rua.

$$p^{2}-4p-12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei p^{2}+ap+bp-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\left(-12\right)=-12$$

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 -12.

$$1,-12 2,-6 3,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-12=-11 2-6=-4 3-4=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=-6 b=2$$

Tuhia anō te p^{2}-4p-12 hei \left(p^{2}-6p\right)+\left(2p-12\right).

$$\left(p^{2}-6p\right)+\left(2p-12\right)$$

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

$$p\left(p-6\right)+2\left(p-6\right)$$

Whakatauwehea atu te kīanga pātahi p-6 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(p-6\right)\left(p+2\right)$$

Hei kimi otinga whārite, me whakaoti te p-6=0 me te p+2=0.

$$p=6 p=-2$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 4p mai i ngā taha e rua.

$$p^{2}-4p=12$$

Tangohia te 12 mai i ngā taha e rua.

$$p^{2}-4p-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -4 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$p=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-12\right)}}{2}$$

Pūrua -4.

$$p=\frac{-\left(-4\right)±\sqrt{16-4\left(-12\right)}}{2}$$

Whakareatia -4 ki te -12.

$$p=\frac{-\left(-4\right)±\sqrt{16+48}}{2}$$

Tāpiri 16 ki te 48.

$$p=\frac{-\left(-4\right)±\sqrt{64}}{2}$$

Tuhia te pūtakerua o te 64.

$$p=\frac{-\left(-4\right)±8}{2}$$

Ko te tauaro o -4 ko 4.

$$p=\frac{4±8}{2}$$

Nā, me whakaoti te whārite p=\frac{4±8}{2} ina he tāpiri te ±. Tāpiri 4 ki te 8.

$$p=\frac{12}{2}$$

Whakawehe 12 ki te 2.

$$p=6$$

Nā, me whakaoti te whārite p=\frac{4±8}{2} ina he tango te ±. Tango 8 mai i 4.

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

Whakawehe -4 ki te 2.

$$p=-2$$

Kua oti te whārite te whakatau.

$$p=6 p=-2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 4p mai i ngā taha e rua.

$$p^{2}-4p=12$$

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

$$p^{2}-4p+\left(-2\right)^{2}=12+\left(-2\right)^{2}$$

Pūrua -2.

$$p^{2}-4p+4=12+4$$

Tāpiri 12 ki te 4.

$$p^{2}-4p+4=16$$

Tauwehea p^{2}-4p+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}.

$$\left(p-2\right)^{2}=16$$

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

$$\sqrt{\left(p-2\right)^{2}}=\sqrt{16}$$

Whakarūnātia.

$$p-2=4 p-2=-4$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$p=6 p=-2$$