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

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

Whakaoti mō x

$$x=-2$$

$$x=3$$

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+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=-6=-6$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -6.

$$-1,6 -2,3$$

Tātaihia te tapeke mō ia takirua.

$$-1+6=5 -2+3=1$$

Ko te otinga te takirua ka hoatu i te tapeke 1.

$$a=3 b=-2$$

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

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

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

$$-x\left(x-3\right)-2\left(x-3\right)$$

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

$$\left(x-3\right)\left(-x-2\right)$$

Hei kimi otinga whārite, me whakaoti te x-3=0 me te -x-2=0.

$$x=3 x=-2$$

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.

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

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

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

Pūrua 1.

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

Whakareatia -4 ki te -1.

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

Whakareatia 4 ki te 6.

$$x=\frac{-1±\sqrt{1+24}}{2\left(-1\right)}$$

Tāpiri 1 ki te 24.

$$x=\frac{-1±\sqrt{25}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 25.

$$x=\frac{-1±5}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-1±5}{-2}$$

Nā, me whakaoti te whārite x=\frac{-1±5}{-2} ina he tāpiri te ±. Tāpiri -1 ki te 5.

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

Whakawehe 4 ki te -2.

$$x=-2$$

Nā, me whakaoti te whārite x=\frac{-1±5}{-2} ina he tango te ±. Tango 5 mai i -1.

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

Whakawehe -6 ki te -2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=-2 x=3$$

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.

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

Me tango 6 mai i ngā taha e rua o te whārite.

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

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

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

Whakawehea ngā taha e rua ki te -1.

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

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

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

Whakawehe 1 ki te -1.

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

Whakawehe -6 ki te -1.

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

Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=6+\left(-\frac{1}{2}\right)^{2}$$

Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-x+\frac{1}{4}=6+\frac{1}{4}$$

Tāpiri 6 ki te \frac{1}{4}.

$$x^{2}-x+\frac{1}{4}=\frac{25}{4}$$

Tauwehea x^{2}-x+\frac{1}{4}. 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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{25}{4}$$

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

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Whakarūnātia.

$$x-\frac{1}{2}=\frac{5}{2} x-\frac{1}{2}=-\frac{5}{2}$$

Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.

$$x=3 x=-2$$