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

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

Whakaoti mō x

$$x=-7$$

$$x=3$$

Ngā Upane Whakamahi Tauwehe

Hei whakaoti i te whārite, whakatauwehea te x^{2}+4x-21 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.

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

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 -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.

$$a=-3 b=7$$

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

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

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

$$x=3 x=-7$$

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-21. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=1\left(-21\right)=-21$$

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 -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}.

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

Pūrua 4.

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

Whakareatia -4 ki te -21.

$$x=\frac{-4±\sqrt{16+84}}{2}$$

Tāpiri 16 ki te 84.

$$x=\frac{-4±\sqrt{100}}{2}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-4±10}{2}$$

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

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

Whakawehe 6 ki te 2.

$$x=3$$

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

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

Whakawehe -14 ki te 2.

$$x=-7$$

Kua oti te whārite te whakatau.

$$x=3 x=-7$$

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 tāpiri 21 ki ngā taha e rua o te whārite.

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

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

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

Tango -21 mai i 0.

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

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 te x^{2}+4x+4. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe 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.

$$x=3 x=-7$$