Whakaoti i te x^2-3x-108=0

$$x ^ { 2 } - 3 x - 108 = 0$$

Whakaoti mō x

$$x=-9$$

$$x=12$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-3x-108 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.

$$a+b=-3 ab=-108$$

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 -108.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-12 b=9$$

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

$$\left(x-12\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+9=0.

$$x=12 x=-9$$

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-108. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-3 ab=1\left(-108\right)=-108$$

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 -108.

$$1,-108 2,-54 3,-36 4,-27 6,-18 9,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-108=-107 2-54=-52 3-36=-33 4-27=-23 6-18=-12 9-12=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -3.

$$a=-12 b=9$$

Tuhia anō te x^{2}-3x-108 hei \left(x^{2}-12x\right)+\left(9x-108\right).

$$\left(x^{2}-12x\right)+\left(9x-108\right)$$

Tauwehea te x i te tuatahi me te 9 i te rōpū tuarua.

$$x\left(x-12\right)+9\left(x-12\right)$$

Whakatauwehea atu te kīanga pātahi x-12 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-12\right)\left(x+9\right)$$

Hei kimi otinga whārite, me whakaoti te x-12=0 me te x+9=0.

$$x=12 x=-9$$

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.

$$x^{2}-3x-108=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -3 mō b, me -108 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-108\right)}}{2}$$

Pūrua -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-108\right)}}{2}$$

Whakareatia -4 ki te -108.

$$x=\frac{-\left(-3\right)±\sqrt{9+432}}{2}$$

Tāpiri 9 ki te 432.

$$x=\frac{-\left(-3\right)±\sqrt{441}}{2}$$

Tuhia te pūtakerua o te 441.

$$x=\frac{-\left(-3\right)±21}{2}$$

Ko te tauaro o -3 ko 3.

$$x=\frac{3±21}{2}$$

Nā, me whakaoti te whārite x=\frac{3±21}{2} ina he tāpiri te ±. Tāpiri 3 ki te 21.

$$x=\frac{24}{2}$$

Whakawehe 24 ki te 2.

$$x=12$$

Nā, me whakaoti te whārite x=\frac{3±21}{2} ina he tango te ±. Tango 21 mai i 3.

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

Whakawehe -18 ki te 2.

$$x=-9$$

Kua oti te whārite te whakatau.

$$x=12 x=-9$$

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.

$$x^{2}-3x-108=0$$

Me tāpiri 108 ki ngā taha e rua o te whārite.

$$x^{2}-3x-108-\left(-108\right)=-\left(-108\right)$$

Mā te tango i te -108 i a ia ake anō ka toe ko te 0.

$$x^{2}-3x=-\left(-108\right)$$

Tango -108 mai i 0.

$$x^{2}-3x=108$$

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

$$x^{2}-3x+\left(-\frac{3}{2}\right)^{2}=108+\left(-\frac{3}{2}\right)^{2}$$

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=108+\frac{9}{4}$$

Tāpiri 108 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{441}{4}$$

Tauwehea x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{441}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{441}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{21}{2} x-\frac{3}{2}=-\frac{21}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=12 x=-9$$