Whakaoti i te 2-3x(4-x)=x^2-16

$$2 - 3 x ( 4 - x ) = x ^ { 2 } - 16$$

Whakaoti mō x

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te x^{2} mai i ngā taha e rua.

$$2-3x\left(4-x\right)-x^{2}=-16$$

Me tāpiri te 16 ki ngā taha e rua.

$$2-3x\left(4-x\right)-x^{2}+16=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3x ki te 4-x.

$$2-12x+3x^{2}-x^{2}+16=0$$

Pahekotia te 3x^{2} me -x^{2}, ka 2x^{2}.

$$2-12x+2x^{2}+16=0$$

Tāpirihia te 2 ki te 16, ka 18.

$$18-12x+2x^{2}=0$$

Whakawehea ngā taha e rua ki te 2.

$$9-6x+x^{2}=0$$

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

$$x^{2}-6x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 9=9$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

$$-1,-9 -3,-3$$

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-3 b=-3$$

Tuhia anō te x^{2}-6x+9 hei \left(x^{2}-3x\right)+\left(-3x+9\right).

$$\left(x^{2}-3x\right)+\left(-3x+9\right)$$

Tauwehea te x i te tuatahi me te -3 i te rōpū tuarua.

$$x\left(x-3\right)-3\left(x-3\right)$$

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-3\right)\left(x-3\right)$$

Tuhia anōtia hei pūrua huarua.

$$\left(x-3\right)^{2}$$

Hei kimi i te otinga whārite, whakaotia te x-3=0.

$$x=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te x^{2} mai i ngā taha e rua.

$$2-3x\left(4-x\right)-x^{2}=-16$$

Me tāpiri te 16 ki ngā taha e rua.

$$2-3x\left(4-x\right)-x^{2}+16=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3x ki te 4-x.

$$2-12x+3x^{2}-x^{2}+16=0$$

Pahekotia te 3x^{2} me -x^{2}, ka 2x^{2}.

$$2-12x+2x^{2}+16=0$$

Tāpirihia te 2 ki te 16, ka 18.

$$18-12x+2x^{2}=0$$

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.

$$2x^{2}-12x+18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -12 mō b, me 18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 2\times 18}}{2\times 2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 2\times 18}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-12\right)±\sqrt{144-8\times 18}}{2\times 2}$$

Whakareatia -8 ki te 18.

$$x=\frac{-\left(-12\right)±\sqrt{144-144}}{2\times 2}$$

Tāpiri 144 ki te -144.

$$x=\frac{-\left(-12\right)±\sqrt{0}}{2\times 2}$$

Tuhia te pūtakerua o te 0.

$$x=-\frac{-12}{2\times 2}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{12}{4}$$

Whakawehe 12 ki te 4.

$$x=3$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te x^{2} mai i ngā taha e rua.

$$2-3x\left(4-x\right)-x^{2}=-16$$

Whakamahia te āhuatanga tohatoha hei whakarea te -3x ki te 4-x.

$$2-12x+3x^{2}-x^{2}=-16$$

Pahekotia te 3x^{2} me -x^{2}, ka 2x^{2}.

$$2-12x+2x^{2}=-16$$

Tangohia te 2 mai i ngā taha e rua.

$$-12x+2x^{2}=-16-2$$

Tangohia te 2 i te -16, ka -18.

$$-12x+2x^{2}=-18$$

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.

$$2x^{2}-12x=-18$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-12x}{2}=-\frac{18}{2}$$

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

$$x^{2}+\left(-\frac{12}{2}\right)x=-\frac{18}{2}$$

Whakawehe -12 ki te 2.

$$x^{2}-6x=-\frac{18}{2}$$

Whakawehe -18 ki te 2.

$$x^{2}-6x=-9$$

Whakawehea te -6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -3. Nā, tāpiria te pūrua o te -3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-6x+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-9+9$$

Tāpiri -9 ki te 9.

$$x^{2}-6x+9=0$$

Tauwehea x^{2}-6x+9. 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}.

$$\left(x-3\right)^{2}=0$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-3=0 x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$x=3 x=3$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=3$$