Whakaoti i te 2x^2-8x-24=0

$$2 x ^ { 2 } - 8 x - 24 = 0$$

Whakaoti mō x

$$x=-2$$

$$x=6$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-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 x^{2}-4x-12 hei \left(x^{2}-6x\right)+\left(2x-12\right).

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

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

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

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

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

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

$$x=6 x=-2$$

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.

$$2x^{2}-8x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -8 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 2\left(-24\right)}}{2\times 2}$$

Pūrua -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 2\left(-24\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-8\right)±\sqrt{64-8\left(-24\right)}}{2\times 2}$$

Whakareatia -8 ki te -24.

$$x=\frac{-\left(-8\right)±\sqrt{64+192}}{2\times 2}$$

Tāpiri 64 ki te 192.

$$x=\frac{-\left(-8\right)±\sqrt{256}}{2\times 2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-8\right)±16}{2\times 2}$$

Ko te tauaro o -8 ko 8.

$$x=\frac{8±16}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{8±16}{4}$$

Nā, me whakaoti te whārite x=\frac{8±16}{4} ina he tāpiri te ±. Tāpiri 8 ki te 16.

$$x=\frac{24}{4}$$

Whakawehe 24 ki te 4.

$$x=6$$

Nā, me whakaoti te whārite x=\frac{8±16}{4} ina he tango te ±. Tango 16 mai i 8.

$$x=-\frac{8}{4}$$

Whakawehe -8 ki te 4.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=6 x=-2$$

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.

$$2x^{2}-8x-24=0$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$2x^{2}-8x-24-\left(-24\right)=-\left(-24\right)$$

Mā te tango i te -24 i a ia ake anō ka toe ko te 0.

$$2x^{2}-8x=-\left(-24\right)$$

Tango -24 mai i 0.

$$2x^{2}-8x=24$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-8x}{2}=\frac{24}{2}$$

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

$$x^{2}+\left(-\frac{8}{2}\right)x=\frac{24}{2}$$

Whakawehe -8 ki te 2.

$$x^{2}-4x=\frac{24}{2}$$

Whakawehe 24 ki te 2.

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

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.

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

Pūrua -2.

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

Tāpiri 12 ki te 4.

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

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}.

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

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

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

Whakarūnātia.

$$x-2=4 x-2=-4$$

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

$$x=6 x=-2$$