Whakaoti i te x^2-6x=-9

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

Whakaoti mō x

$$x=3$$

Ngā Upane Whakamahi Tauwehe

Me tāpiri te 9 ki ngā taha e rua.

$$x^{2}-6x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te x^{2}-6x+9 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.

$$a+b=-6 ab=9$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

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

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

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

$$a=-3 b=-3$$

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

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

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te x-3=0.

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Me tāpiri te 9 ki ngā taha e rua.

$$x^{2}-6x+9=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei x^{2}+ax+bx+9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 9=9$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 9.

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

Tātaihia te tapeke mō ia takirua.

$$-1-9=-10 -3-3=-6$$

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

$$a=-3 b=-3$$

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

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

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

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

Whakatauwehea atu te kīanga pātahi x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Tuhia anōtia hei pūrua huarua.

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

Hei kimi i te otinga whārite, whakaotia te x-3=0.

$$x=3$$

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.

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

Me tāpiri 9 ki ngā taha e rua o te whārite.

$$x^{2}-6x-\left(-9\right)=-9-\left(-9\right)$$

Mā te tango i te -9 i a ia ake anō ka toe ko te 0.

$$x^{2}-6x-\left(-9\right)=0$$

Tango -9 mai i 0.

$$x^{2}-6x+9=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me 9 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 9}}{2}$$

Whakareatia -4 ki te 9.

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

Tāpiri 36 ki te -36.

$$x=\frac{-\left(-6\right)±\sqrt{0}}{2}$$

Tuhia te pūtakerua o te 0.

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

Ko te tauaro o -6 ko 6.

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

Whakawehe 6 ki te 2.

$$x=3$$

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.

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

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.

$$x^{2}-6x+\left(-3\right)^{2}=-9+\left(-3\right)^{2}$$

Pūrua -3.

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

Tāpiri -9 ki te 9.

$$x^{2}-6x+9=0$$

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}.

$$\left(x-3\right)^{2}=0$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{0}$$

Whakarūnātia.

$$x-3=0 x-3=0$$

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

$$x=3 x=3$$

Kua oti te whārite te whakatau. He ōrite ngā whakatau.

$$x=3$$