Whakaoti i te 6x-x^2=8

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

Whakaoti mō x

$$x=2$$

$$x=4$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 8 mai i ngā taha e rua.

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

Hurinahatia te pūrau ki te āhua tānga ngahuru. Whakaraupapahia ngā kīanga tau mai i te pū teitei rawa ki te mea iti rawa.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=6 ab=-\left(-8\right)=8$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga 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.

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

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.

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

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}.

$$x=\frac{-6±\sqrt{6^{2}-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\left(-1\right)\left(-8\right)}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-6±\sqrt{36+4\left(-8\right)}}{2\left(-1\right)}$$

Whakareatia 4 ki te -8.

$$x=\frac{-6±\sqrt{36-32}}{2\left(-1\right)}$$

Tāpiri 36 ki te -32.

$$x=\frac{-6±\sqrt{4}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 4.

$$x=\frac{-6±2}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-6±2}{-2}$$

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

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

Whakawehe -4 ki te -2.

$$x=2$$

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

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

Whakawehe -8 ki te -2.

$$x=4$$

Kua oti te whārite te whakatau.

$$x=2 x=4$$

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.

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

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}+6x}{-1}=\frac{8}{-1}$$

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

$$x^{2}+\frac{6}{-1}x=\frac{8}{-1}$$

Whakawehe 6 ki te -1.

$$x^{2}-6x=\frac{8}{-1}$$

Whakawehe 8 ki te -1.

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

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.

$$x=4 x=2$$