Whakaoti i te 2x^2-2x-12=28

$$2 x ^ { 2 } - 2 x - 12 = 28$$

Whakaoti mō x

$$x=-4$$

$$x=5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 28 mai i ngā taha e rua.

$$2x^{2}-2x-12-28=0$$

Tangohia te 28 i te -12, ka -40.

$$2x^{2}-2x-40=0$$

Whakawehea ngā taha e rua ki te 2.

$$x^{2}-x-20=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-20. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=1\left(-20\right)=-20$$

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 -20.

$$1,-20 2,-10 4,-5$$

Tātaihia te tapeke mō ia takirua.

$$1-20=-19 2-10=-8 4-5=-1$$

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

$$a=-5 b=4$$

Tuhia anō te x^{2}-x-20 hei \left(x^{2}-5x\right)+\left(4x-20\right).

$$\left(x^{2}-5x\right)+\left(4x-20\right)$$

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

$$x\left(x-5\right)+4\left(x-5\right)$$

Whakatauwehea atu te kīanga pātahi x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-5\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-5=0 me te x+4=0.

$$x=5 x=-4$$

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.

$$2x^{2}-2x-12=28$$

Me tango 28 mai i ngā taha e rua o te whārite.

$$2x^{2}-2x-12-28=28-28$$

Mā te tango i te 28 i a ia ake anō ka toe ko te 0.

$$2x^{2}-2x-12-28=0$$

Tango 28 mai i -12.

$$2x^{2}-2x-40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -2 mō b, me -40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\times 2\left(-40\right)}}{2\times 2}$$

Pūrua -2.

$$x=\frac{-\left(-2\right)±\sqrt{4-4\times 2\left(-40\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-2\right)±\sqrt{4-8\left(-40\right)}}{2\times 2}$$

Whakareatia -8 ki te -40.

$$x=\frac{-\left(-2\right)±\sqrt{4+320}}{2\times 2}$$

Tāpiri 4 ki te 320.

$$x=\frac{-\left(-2\right)±\sqrt{324}}{2\times 2}$$

Tuhia te pūtakerua o te 324.

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

Ko te tauaro o -2 ko 2.

$$x=\frac{2±18}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{2±18}{4}$$

Nā, me whakaoti te whārite x=\frac{2±18}{4} ina he tāpiri te ±. Tāpiri 2 ki te 18.

$$x=\frac{20}{4}$$

Whakawehe 20 ki te 4.

$$x=5$$

Nā, me whakaoti te whārite x=\frac{2±18}{4} ina he tango te ±. Tango 18 mai i 2.

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

Whakawehe -16 ki te 4.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=5 x=-4$$

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.

$$2x^{2}-2x-12=28$$

Me tāpiri 12 ki ngā taha e rua o te whārite.

$$2x^{2}-2x-12-\left(-12\right)=28-\left(-12\right)$$

Mā te tango i te -12 i a ia ake anō ka toe ko te 0.

$$2x^{2}-2x=28-\left(-12\right)$$

Tango -12 mai i 28.

$$2x^{2}-2x=40$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-2x}{2}=\frac{40}{2}$$

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

$$x^{2}+\left(-\frac{2}{2}\right)x=\frac{40}{2}$$

Whakawehe -2 ki te 2.

$$x^{2}-x=\frac{40}{2}$$

Whakawehe 40 ki te 2.

$$x^{2}-x=20$$

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=20+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=20+\frac{1}{4}$$

Tāpiri 20 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{81}{4}$$

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

$$\left(x-\frac{1}{2}\right)^{2}=\frac{81}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{9}{2} x-\frac{1}{2}=-\frac{9}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=5 x=-4$$