Whakaoti i te x^2-4x-32=0

$$x ^ { 2 } - 4 x - 32 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=8$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-4x-32 mā te whakamahi i te tātai x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=-32$$

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 -32.

$$1,-32 2,-16 4,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-32=-31 2-16=-14 4-8=-4$$

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

$$a=-8 b=4$$

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

$$\left(x-8\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+4=0.

$$x=8 x=-4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-32. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=1\left(-32\right)=-32$$

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 -32.

$$1,-32 2,-16 4,-8$$

Tātaihia te tapeke mō ia takirua.

$$1-32=-31 2-16=-14 4-8=-4$$

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

$$a=-8 b=4$$

Tuhia anō te x^{2}-4x-32 hei \left(x^{2}-8x\right)+\left(4x-32\right).

$$\left(x^{2}-8x\right)+\left(4x-32\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(x-8\right)+4\left(x-8\right)$$

Whakatauwehea atu te kīanga pātahi x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-8\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-8=0 me te x+4=0.

$$x=8 x=-4$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$x^{2}-4x-32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -4 mō b, me -32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-32\right)}}{2}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-32\right)}}{2}$$

Whakareatia -4 ki te -32.

$$x=\frac{-\left(-4\right)±\sqrt{16+128}}{2}$$

Tāpiri 16 ki te 128.

$$x=\frac{-\left(-4\right)±\sqrt{144}}{2}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-\left(-4\right)±12}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±12}{2}$$

Nā, me whakaoti te whārite x=\frac{4±12}{2} ina he tāpiri te ±. Tāpiri 4 ki te 12.

$$x=\frac{16}{2}$$

Whakawehe 16 ki te 2.

$$x=8$$

Nā, me whakaoti te whārite x=\frac{4±12}{2} ina he tango te ±. Tango 12 mai i 4.

$$x=-\frac{8}{2}$$

Whakawehe -8 ki te 2.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=8 x=-4$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$x^{2}-4x-32=0$$

Me tāpiri 32 ki ngā taha e rua o te whārite.

$$x^{2}-4x-32-\left(-32\right)=-\left(-32\right)$$

Mā te tango i te -32 i a ia ake anō ka toe ko te 0.

$$x^{2}-4x=-\left(-32\right)$$

Tango -32 mai i 0.

$$x^{2}-4x=32$$

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.

$$x^{2}-4x+\left(-2\right)^{2}=32+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=32+4$$

Tāpiri 32 ki te 4.

$$x^{2}-4x+4=36$$

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

$$\left(x-2\right)^{2}=36$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{36}$$

Whakarūnātia.

$$x-2=6 x-2=-6$$

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

$$x=8 x=-4$$