Whakaoti i te 3x^2+6x-24=0

$$3 x ^ { 2 } + 6 x - 24 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=2$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 3.

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

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.

$$a+b=2 ab=1\left(-8\right)=-8$$

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 -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

Ko te otinga te takirua ka hoatu i te tapeke 2.

$$a=-2 b=4$$

Tuhia anō te x^{2}+2x-8 hei \left(x^{2}-2x\right)+\left(4x-8\right).

$$\left(x^{2}-2x\right)+\left(4x-8\right)$$

Tauwehea te x i te tuatahi me te 4 i te rōpū tuarua.

$$x\left(x-2\right)+4\left(x-2\right)$$

Whakatauwehea atu te kīanga pātahi x-2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-2\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te x-2=0 me te x+4=0.

$$x=2 x=-4$$

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.

$$3x^{2}+6x-24=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 6 mō b, me -24 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 3\left(-24\right)}}{2\times 3}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 3\left(-24\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-6±\sqrt{36-12\left(-24\right)}}{2\times 3}$$

Whakareatia -12 ki te -24.

$$x=\frac{-6±\sqrt{36+288}}{2\times 3}$$

Tāpiri 36 ki te 288.

$$x=\frac{-6±\sqrt{324}}{2\times 3}$$

Tuhia te pūtakerua o te 324.

$$x=\frac{-6±18}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-6±18}{6}$$

Nā, me whakaoti te whārite x=\frac{-6±18}{6} ina he tāpiri te ±. Tāpiri -6 ki te 18.

$$x=\frac{12}{6}$$

Whakawehe 12 ki te 6.

$$x=2$$

Nā, me whakaoti te whārite x=\frac{-6±18}{6} ina he tango te ±. Tango 18 mai i -6.

$$x=-\frac{24}{6}$$

Whakawehe -24 ki te 6.

$$x=-4$$

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.

$$3x^{2}+6x-24=0$$

Me tāpiri 24 ki ngā taha e rua o te whārite.

$$3x^{2}+6x-24-\left(-24\right)=-\left(-24\right)$$

Mā te tango i te -24 i a ia ake anō ka toe ko te 0.

$$3x^{2}+6x=-\left(-24\right)$$

Tango -24 mai i 0.

$$3x^{2}+6x=24$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+6x}{3}=\frac{24}{3}$$

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

$$x^{2}+\frac{6}{3}x=\frac{24}{3}$$

Whakawehe 6 ki te 3.

$$x^{2}+2x=\frac{24}{3}$$

Whakawehe 24 ki te 3.

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

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

$$x^{2}+2x+1^{2}=8+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=8+1$$

Tāpiri 8 ki te 1.

$$x^{2}+2x+1=9$$

Tauwehea x^{2}+2x+1. 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}.

$$\left(x+1\right)^{2}=9$$

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

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{9}$$

Whakarūnātia.

$$x+1=3 x+1=-3$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=2 x=-4$$