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

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

Whakaoti mō x

$$x=-3$$

$$x=7$$

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 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.

$$a=-7 b=3$$

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

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

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

$$x=7 x=-3$$

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-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 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.

$$a=-7 b=3$$

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

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

Tauwehea te x i te tuatahi me te 3 i te rōpū tuarua.

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

Whakatauwehea atu te kīanga pātahi x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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

$$x=7 x=-3$$

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}.

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

Pūrua -4.

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

Whakareatia -4 ki te -21.

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

Tāpiri 16 ki te 84.

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

Tuhia te pūtakerua o te 100.

$$x=\frac{-\left(-4\right)±10}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±10}{2}$$

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 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.

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

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 tāpiri 2 ki ngā taha e rua o te whārite.

$$x=7 x=-3$$