Whakaoti i te x^2-6x+8=0

$${ x }^{ 2 } -6x+8=0$$

Whakaoti mō x

$$x=2$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}-6x+8 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.

$$a+b=-6 ab=8$$

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 8.

$$-1,-8 -2,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-8=-9 -2-4=-6$$

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

$$a=-4 b=-2$$

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

$$\left(x-4\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x-2=0.

$$x=4 x=2$$

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+8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=1\times 8=8$$

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 8.

$$-1,-8 -2,-4$$

Tātaihia te tapeke mō ia takirua.

$$-1-8=-9 -2-4=-6$$

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

$$a=-4 b=-2$$

Tuhia anō te x^{2}-6x+8 hei \left(x^{2}-4x\right)+\left(-2x+8\right).

$$\left(x^{2}-4x\right)+\left(-2x+8\right)$$

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

$$x\left(x-4\right)-2\left(x-4\right)$$

Whakatauwehea atu te kīanga pātahi x-4 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-4\right)\left(x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-4=0 me te x-2=0.

$$x=4 x=2$$

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.

$$x^{2}-6x+8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -6 mō b, me 8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 8}}{2}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 8}}{2}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-6\right)±\sqrt{36-32}}{2}$$

Tāpiri 36 ki te -32.

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

Tuhia te pūtakerua o te 4.

$$x=\frac{-\left(-6\right)±2}{2}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±2}{2}$$

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

$$x=\frac{8}{2}$$

Whakawehe 8 ki te 2.

$$x=4$$

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

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

Whakawehe 4 ki te 2.

$$x=2$$

Kua oti te whārite te whakatau.

$$x=4 x=2$$

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.

$$x^{2}-6x+8=0$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$x^{2}-6x+8-8=-8$$

Mā te tango i te 8 i a ia ake anō ka toe ko te 0.

$$x^{2}-6x=-8$$

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.

$$x^{2}-6x+\left(-3\right)^{2}=-8+\left(-3\right)^{2}$$

Pūrua -3.

$$x^{2}-6x+9=-8+9$$

Tāpiri -8 ki te 9.

$$x^{2}-6x+9=1$$

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}.

$$\left(x-3\right)^{2}=1$$

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

$$\sqrt{\left(x-3\right)^{2}}=\sqrt{1}$$

Whakarūnātia.

$$x-3=1 x-3=-1$$

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

$$x=4 x=2$$