Whakaoti i te x^2-6x=27

$$x ^ { 2 } - 6 x = 27$$

Whakaoti mō x

$$x=-3$$

$$x=9$$

Ngā Upane Whakamahi Tauwehe

Tangohia te 27 mai i ngā taha e rua.

$$x^{2}-6x-27=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-6x-27 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.

$$a+b=-6 ab=-27$$

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 -27.

$$1,-27 3,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-27=-26 3-9=-6$$

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

$$a=-9 b=3$$

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

$$\left(x-9\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+3=0.

$$x=9 x=-3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 27 mai i ngā taha e rua.

$$x^{2}-6x-27=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx-27. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\left(-27\right)=-27$$

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 -27.

$$1,-27 3,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-27=-26 3-9=-6$$

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

$$a=-9 b=3$$

Tuhia anō te x^{2}-6x-27 hei \left(x^{2}-9x\right)+\left(3x-27\right).

$$\left(x^{2}-9x\right)+\left(3x-27\right)$$

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

$$x\left(x-9\right)+3\left(x-9\right)$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(x+3\right)$$

Hei kimi otinga whārite, me whakaoti te x-9=0 me te x+3=0.

$$x=9 x=-3$$

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.

$$x^{2}-6x=27$$

Me tango 27 mai i ngā taha e rua o te whārite.

$$x^{2}-6x-27=27-27$$

Mā te tango i te 27 i a ia ake anō ka toe ko te 0.

$$x^{2}-6x-27=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me -27 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\left(-27\right)}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\left(-27\right)}}{2}$$

Whakareatia -4 ki te -27.

$$x=\frac{-\left(-6\right)±\sqrt{36+108}}{2}$$

Tāpiri 36 ki te 108.

$$x=\frac{-\left(-6\right)±\sqrt{144}}{2}$$

Tuhia te pūtakerua o te 144.

$$x=\frac{-\left(-6\right)±12}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±12}{2}$$

Nā, me whakaoti te whārite x=\frac{6±12}{2} ina he tāpiri te ±. Tāpiri 6 ki te 12.

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

Whakawehe 18 ki te 2.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{6±12}{2} ina he tango te ±. Tango 12 mai i 6.

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

Whakawehe -6 ki te 2.

$$x=-3$$

Kua oti te whārite te whakatau.

$$x=9 x=-3$$

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.

$$x^{2}-6x=27$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=27+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=27+9$$

Tāpiri 27 ki te 9.

$$x^{2}-6x+9=36$$

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}.

$$\left(x-3\right)^{2}=36$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{36}$$

Whakarūnātia.

$$x-3=6 x-3=-6$$

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

$$x=9 x=-3$$