Whakaoti i te 2x^2-x-36=0

$$2 x ^ { 2 } - x - 36 = 0$$

Whakaoti mō x

$$x=-4$$

$$x = \frac{9}{2} = 4\frac{1}{2} = 4.5$$

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 2x^{2}+ax+bx-36. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=2\left(-36\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-9 b=8$$

Tuhia anō te 2x^{2}-x-36 hei \left(2x^{2}-9x\right)+\left(8x-36\right).

$$\left(2x^{2}-9x\right)+\left(8x-36\right)$$

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

$$x\left(2x-9\right)+4\left(2x-9\right)$$

Whakatauwehea atu te kīanga pātahi 2x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-9\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te 2x-9=0 me te x+4=0.

$$x=\frac{9}{2} x=-4$$

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.

$$2x^{2}-x-36=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -1 mō b, me -36 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 2\left(-36\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-1\right)±\sqrt{1-8\left(-36\right)}}{2\times 2}$$

Whakareatia -8 ki te -36.

$$x=\frac{-\left(-1\right)±\sqrt{1+288}}{2\times 2}$$

Tāpiri 1 ki te 288.

$$x=\frac{-\left(-1\right)±\sqrt{289}}{2\times 2}$$

Tuhia te pūtakerua o te 289.

$$x=\frac{-\left(-1\right)±17}{2\times 2}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±17}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{1±17}{4}$$

Nā, me whakaoti te whārite x=\frac{1±17}{4} ina he tāpiri te ±. Tāpiri 1 ki te 17.

$$x=\frac{18}{4}$$

Whakahekea te hautanga \frac{18}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{9}{2}$$

Nā, me whakaoti te whārite x=\frac{1±17}{4} ina he tango te ±. Tango 17 mai i 1.

$$x=-\frac{16}{4}$$

Whakawehe -16 ki te 4.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=\frac{9}{2} x=-4$$

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.

$$2x^{2}-x-36=0$$

Me tāpiri 36 ki ngā taha e rua o te whārite.

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

Mā te tango i te -36 i a ia ake anō ka toe ko te 0.

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

Tango -36 mai i 0.

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

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-x}{2}=\frac{36}{2}$$

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

$$x^{2}-\frac{1}{2}x=\frac{36}{2}$$

Whakawehe 36 ki te 2.

$$x^{2}-\frac{1}{2}x=18$$

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{1}{2}x+\left(-\frac{1}{4}\right)^{2}=18+\left(-\frac{1}{4}\right)^{2}$$

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=18+\frac{1}{16}$$

Tāpiri 18 ki te \frac{1}{16}.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{289}{16}$$

Tauwehea te x^{2}-\frac{1}{2}x+\frac{1}{16}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{289}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{289}{16}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{17}{4} x-\frac{1}{4}=-\frac{17}{4}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{9}{2} x=-4$$