Whakaoti i te 3x^2+23x-8=0

$$3 x ^ { 2 } + 23 x - 8 = 0$$

Whakaoti mō x

$$x=-8$$

$$x=\frac{1}{3}\approx 0.333333333$$

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 3x^{2}+ax+bx-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=23 ab=3\left(-8\right)=-24$$

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 -24.

$$-1,24 -2,12 -3,8 -4,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

Ko te otinga te takirua ka hoatu i te tapeke 23.

$$a=-1 b=24$$

Tuhia anō te 3x^{2}+23x-8 hei \left(3x^{2}-x\right)+\left(24x-8\right).

$$\left(3x^{2}-x\right)+\left(24x-8\right)$$

Tauwehea te x i te tuatahi me te 8 i te rōpū tuarua.

$$x\left(3x-1\right)+8\left(3x-1\right)$$

Whakatauwehea atu te kīanga pātahi 3x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-1\right)\left(x+8\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-1=0 me te x+8=0.

$$x=\frac{1}{3} x=-8$$

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.

$$3x^{2}+23x-8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 23 mō b, me -8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-23±\sqrt{23^{2}-4\times 3\left(-8\right)}}{2\times 3}$$

Pūrua 23.

$$x=\frac{-23±\sqrt{529-4\times 3\left(-8\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-23±\sqrt{529-12\left(-8\right)}}{2\times 3}$$

Whakareatia -12 ki te -8.

$$x=\frac{-23±\sqrt{529+96}}{2\times 3}$$

Tāpiri 529 ki te 96.

$$x=\frac{-23±\sqrt{625}}{2\times 3}$$

Tuhia te pūtakerua o te 625.

$$x=\frac{-23±25}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-23±25}{6}$$

Nā, me whakaoti te whārite x=\frac{-23±25}{6} ina he tāpiri te ±. Tāpiri -23 ki te 25.

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

Whakahekea te hautanga \frac{2}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{3}$$

Nā, me whakaoti te whārite x=\frac{-23±25}{6} ina he tango te ±. Tango 25 mai i -23.

$$x=-\frac{48}{6}$$

Whakawehe -48 ki te 6.

$$x=-8$$

Kua oti te whārite te whakatau.

$$x=\frac{1}{3} x=-8$$

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.

$$3x^{2}+23x-8=0$$

Me tāpiri 8 ki ngā taha e rua o te whārite.

$$3x^{2}+23x-8-\left(-8\right)=-\left(-8\right)$$

Mā te tango i te -8 i a ia ake anō ka toe ko te 0.

$$3x^{2}+23x=-\left(-8\right)$$

Tango -8 mai i 0.

$$3x^{2}+23x=8$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+23x}{3}=\frac{8}{3}$$

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

$$x^{2}+\frac{23}{3}x=\frac{8}{3}$$

Whakawehea te \frac{23}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{23}{6}. Nā, tāpiria te pūrua o te \frac{23}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{23}{3}x+\left(\frac{23}{6}\right)^{2}=\frac{8}{3}+\left(\frac{23}{6}\right)^{2}$$

Pūruatia \frac{23}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{23}{3}x+\frac{529}{36}=\frac{8}{3}+\frac{529}{36}$$

Tāpiri \frac{8}{3} ki te \frac{529}{36} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}+\frac{23}{3}x+\frac{529}{36}=\frac{625}{36}$$

Tauwehea x^{2}+\frac{23}{3}x+\frac{529}{36}. 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}.

$$\left(x+\frac{23}{6}\right)^{2}=\frac{625}{36}$$

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

$$\sqrt{\left(x+\frac{23}{6}\right)^{2}}=\sqrt{\frac{625}{36}}$$

Whakarūnātia.

$$x+\frac{23}{6}=\frac{25}{6} x+\frac{23}{6}=-\frac{25}{6}$$

Me tango \frac{23}{6} mai i ngā taha e rua o te whārite.

$$x=\frac{1}{3} x=-8$$