Whakaoti i te x^2-6x=91

$$x ^ { 2 } - 6 x = 91$$

Whakaoti mō x

$$x=-7$$

$$x=13$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 91 mai i ngā taha e rua.

$$x^{2}-6x-91=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-6x-91 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.

$$a+b=-6 ab=-91$$

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 -91.

$$1,-91 7,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-91=-90 7-13=-6$$

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

$$a=-13 b=7$$

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

$$\left(x-13\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x+7=0.

$$x=13 x=-7$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 91 mai i ngā taha e rua.

$$x^{2}-6x-91=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-91. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\left(-91\right)=-91$$

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 -91.

$$1,-91 7,-13$$

Tātaihia te tapeke mō ia takirua.

$$1-91=-90 7-13=-6$$

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

$$a=-13 b=7$$

Tuhia anō te x^{2}-6x-91 hei \left(x^{2}-13x\right)+\left(7x-91\right).

$$\left(x^{2}-13x\right)+\left(7x-91\right)$$

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(x-13\right)+7\left(x-13\right)$$

Whakatauwehea atu te kīanga pātahi x-13 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-13\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te x-13=0 me te x+7=0.

$$x=13 x=-7$$

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.

$$x^{2}-6x=91$$

Me tango 91 mai i ngā taha e rua o te whārite.

$$x^{2}-6x-91=91-91$$

Mā te tango i te 91 i a ia ake anō ka toe ko te 0.

$$x^{2}-6x-91=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me -91 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-91\right)}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-91\right)}}{2}$$

Whakareatia -4 ki te -91.

$$x=\frac{-\left(-6\right)±\sqrt{36+364}}{2}$$

Tāpiri 36 ki te 364.

$$x=\frac{-\left(-6\right)±\sqrt{400}}{2}$$

Tuhia te pūtakerua o te 400.

$$x=\frac{-\left(-6\right)±20}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±20}{2}$$

Nā, me whakaoti te whārite x=\frac{6±20}{2} ina he tāpiri te ±. Tāpiri 6 ki te 20.

$$x=\frac{26}{2}$$

Whakawehe 26 ki te 2.

$$x=13$$

Nā, me whakaoti te whārite x=\frac{6±20}{2} ina he tango te ±. Tango 20 mai i 6.

$$x=-\frac{14}{2}$$

Whakawehe -14 ki te 2.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=13 x=-7$$

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.

$$x^{2}-6x=91$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=91+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=91+9$$

Tāpiri 91 ki te 9.

$$x^{2}-6x+9=100$$

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}.

$$\left(x-3\right)^{2}=100$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{100}$$

Whakarūnātia.

$$x-3=10 x-3=-10$$

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

$$x=13 x=-7$$