Whakaoti i te 21-4x-x^2=0

$$21 - 4 x - x ^ { 2 } = 0$$

Whakaoti mō x

$$x=-7$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

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

$$-x^{2}-4x+21=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei -x^{2}+ax+bx+21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-4 ab=-21=-21$$

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 -21.

$$1,-21 3,-7$$

Tātaihia te tapeke mō ia takirua.

$$1-21=-20 3-7=-4$$

Ko te otinga te takirua ka hoatu i te tapeke -4.

$$a=3 b=-7$$

Tuhia anō te -x^{2}-4x+21 hei \left(-x^{2}+3x\right)+\left(-7x+21\right).

$$\left(-x^{2}+3x\right)+\left(-7x+21\right)$$

Tauwehea te x i te tuatahi me te 7 i te rōpū tuarua.

$$x\left(-x+3\right)+7\left(-x+3\right)$$

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

$$\left(-x+3\right)\left(x+7\right)$$

Hei kimi otinga whārite, me whakaoti te -x+3=0 me te x+7=0.

$$x=3 x=-7$$

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.

$$-x^{2}-4x+21=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, -4 mō b, me 21 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\left(-1\right)\times 21}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+4\times 21}}{2\left(-1\right)}$$

Whakareatia 4 ki te 21.

$$x=\frac{-\left(-4\right)±\sqrt{16+84}}{2\left(-1\right)}$$

Tāpiri 16 ki te 84.

$$x=\frac{-\left(-4\right)±\sqrt{100}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-4\right)±10}{2\left(-1\right)}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±10}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{4±10}{-2}$$

Nā, me whakaoti te whārite x=\frac{4±10}{-2} ina he tāpiri te ±. Tāpiri 4 ki te 10.

$$x=\frac{14}{-2}$$

Whakawehe 14 ki te -2.

$$x=-7$$

Nā, me whakaoti te whārite x=\frac{4±10}{-2} ina he tango te ±. Tango 10 mai i 4.

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

Whakawehe -6 ki te -2.

$$x=3$$

Kua oti te whārite te whakatau.

$$x=-7 x=3$$

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.

$$-x^{2}-4x+21=0$$

Me tango 21 mai i ngā taha e rua o te whārite.

$$-x^{2}-4x+21-21=-21$$

Mā te tango i te 21 i a ia ake anō ka toe ko te 0.

$$-x^{2}-4x=-21$$

Whakawehea ngā taha e rua ki te -1.

$$\frac{-x^{2}-4x}{-1}=-\frac{21}{-1}$$

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

$$x^{2}+\left(-\frac{4}{-1}\right)x=-\frac{21}{-1}$$

Whakawehe -4 ki te -1.

$$x^{2}+4x=-\frac{21}{-1}$$

Whakawehe -21 ki te -1.

$$x^{2}+4x=21$$

Whakawehea te 4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 2. Nā, tāpiria te pūrua o te 2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+4x+2^{2}=21+2^{2}$$

Pūrua 2.

$$x^{2}+4x+4=21+4$$

Tāpiri 21 ki te 4.

$$x^{2}+4x+4=25$$

Tauwehea x^{2}+4x+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}.

$$\left(x+2\right)^{2}=25$$

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

$$\sqrt{\left(x+2\right)^{2}}=\sqrt{25}$$

Whakarūnātia.

$$x+2=5 x+2=-5$$

Me tango 2 mai i ngā taha e rua o te whārite.

$$x=3 x=-7$$