Whakaoti i te x^2-12x-28=0

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

Whakaoti mō x

$$x=-2$$

$$x=14$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-12x-28 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.

$$a+b=-12 ab=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-28=-27 2-14=-12 4-7=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-14 b=2$$

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

$$\left(x-14\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-14=0 me te x+2=0.

$$x=14 x=-2$$

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-28. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-12 ab=1\left(-28\right)=-28$$

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 -28.

$$1,-28 2,-14 4,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-28=-27 2-14=-12 4-7=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-14 b=2$$

Tuhia anō te x^{2}-12x-28 hei \left(x^{2}-14x\right)+\left(2x-28\right).

$$\left(x^{2}-14x\right)+\left(2x-28\right)$$

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

$$x\left(x-14\right)+2\left(x-14\right)$$

Whakatauwehea atu te kīanga pātahi x-14 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-14\right)\left(x+2\right)$$

Hei kimi otinga whārite, me whakaoti te x-14=0 me te x+2=0.

$$x=14 x=-2$$

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.

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

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -12 mō b, me -28 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-28\right)}}{2}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-28\right)}}{2}$$

Whakareatia -4 ki te -28.

$$x=\frac{-\left(-12\right)±\sqrt{144+112}}{2}$$

Tāpiri 144 ki te 112.

$$x=\frac{-\left(-12\right)±\sqrt{256}}{2}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-12\right)±16}{2}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±16}{2}$$

Nā, me whakaoti te whārite x=\frac{12±16}{2} ina he tāpiri te ±. Tāpiri 12 ki te 16.

$$x=\frac{28}{2}$$

Whakawehe 28 ki te 2.

$$x=14$$

Nā, me whakaoti te whārite x=\frac{12±16}{2} ina he tango te ±. Tango 16 mai i 12.

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

Whakawehe -4 ki te 2.

$$x=-2$$

Kua oti te whārite te whakatau.

$$x=14 x=-2$$

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.

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

Me tāpiri 28 ki ngā taha e rua o te whārite.

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

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

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

Tango -28 mai i 0.

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

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=28+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=28+36$$

Tāpiri 28 ki te 36.

$$x^{2}-12x+36=64$$

Tauwehea x^{2}-12x+36. 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}.

$$\left(x-6\right)^{2}=64$$

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

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{64}$$

Whakarūnātia.

$$x-6=8 x-6=-8$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=14 x=-2$$