Whakaoti i te x^2-2x-8=0

$$x ^ { 2 } - 2 x - 8 = 0$$

Whakaoti mō x

$$x=-2$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-2x-8 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.

$$a+b=-2 ab=-8$$

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 -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-4 b=2$$

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

$$\left(x-4\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+2=0.

$$x=4 x=-2$$

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-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-2 ab=1\left(-8\right)=-8$$

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 -8.

$$1,-8 2,-4$$

Tātaihia te tapeke mō ia takirua.

$$1-8=-7 2-4=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -2.

$$a=-4 b=2$$

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

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

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

$$x\left(x-4\right)+2\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x+2=0.

$$x=4 x=-2$$

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.

$$x^{2}-2x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -2 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-8\right)}}{2}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\left(-8\right)}}{2}$$

Whakareatia -4 ki te -8.

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

Tāpiri 4 ki te 32.

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

Tuhia te pūtakerua o te 36.

$$x=\frac{-\left(-2\right)±6}{2}$$

Ko te tauaro o -2 ko 2.

$$x=\frac{2±6}{2}$$

Nā, me whakaoti te whārite x=\frac{2±6}{2} ina he tāpiri te ±. Tāpiri 2 ki te 6.

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

Whakawehe 8 ki te 2.

$$x=4$$

Nā, me whakaoti te whārite x=\frac{2±6}{2} ina he tango te ±. Tango 6 mai i 2.

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

Whakawehe -4 ki te 2.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=4 x=-2$$

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.

$$x^{2}-2x-8=0$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$x^{2}-2x-8-\left(-8\right)=-\left(-8\right)$$

Mā te tango i te -8 i a ia ake anō ka toe ko te 0.

$$x^{2}-2x=-\left(-8\right)$$

Tango -8 mai i 0.

$$x^{2}-2x=8$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=8+1$$

Tāpiri 8 ki te 1.

$$x^{2}-2x+1=9$$

Tauwehea x^{2}-2x+1. 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}.

$$\left(x-1\right)^{2}=9$$

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

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x-1=3 x-1=-3$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$x=4 x=-2$$