Whakaoti i te 3x^2+14x+8=0

$$3 x ^ { 2 } + 14 x + 8 = 0$$

Whakaoti mō x

$$x=-4$$

$$x=-\frac{2}{3}\approx -0.666666667$$

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.

$$a+b=14 ab=3\times 8=24$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. 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.

$$1+24=25 2+12=14 3+8=11 4+6=10$$

Ko te otinga te takirua ka hoatu i te tapeke 14.

$$a=2 b=12$$

Tuhia anō te 3x^{2}+14x+8 hei \left(3x^{2}+2x\right)+\left(12x+8\right).

$$\left(3x^{2}+2x\right)+\left(12x+8\right)$$

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

$$x\left(3x+2\right)+4\left(3x+2\right)$$

Whakatauwehea atu te kīanga pātahi 3x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x+2\right)\left(x+4\right)$$

Hei kimi otinga whārite, me whakaoti te 3x+2=0 me te x+4=0.

$$x=-\frac{2}{3} x=-4$$

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.

$$3x^{2}+14x+8=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, 14 mō b, me 8 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 3\times 8}}{2\times 3}$$

Pūrua 14.

$$x=\frac{-14±\sqrt{196-4\times 3\times 8}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-14±\sqrt{196-12\times 8}}{2\times 3}$$

Whakareatia -12 ki te 8.

$$x=\frac{-14±\sqrt{196-96}}{2\times 3}$$

Tāpiri 196 ki te -96.

$$x=\frac{-14±\sqrt{100}}{2\times 3}$$

Tuhia te pūtakerua o te 100.

$$x=\frac{-14±10}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{-14±10}{6}$$

Nā, me whakaoti te whārite x=\frac{-14±10}{6} ina he tāpiri te ±. Tāpiri -14 ki te 10.

$$x=-\frac{4}{6}$$

Whakahekea te hautanga \frac{-4}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{2}{3}$$

Nā, me whakaoti te whārite x=\frac{-14±10}{6} ina he tango te ±. Tango 10 mai i -14.

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

Whakawehe -24 ki te 6.

$$x=-4$$

Kua oti te whārite te whakatau.

$$x=-\frac{2}{3} x=-4$$

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.

$$3x^{2}+14x+8=0$$

Me tango 8 mai i ngā taha e rua o te whārite.

$$3x^{2}+14x+8-8=-8$$

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

$$3x^{2}+14x=-8$$

Whakawehea ngā taha e rua ki te 3.

$$\frac{3x^{2}+14x}{3}=-\frac{8}{3}$$

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

$$x^{2}+\frac{14}{3}x=-\frac{8}{3}$$

Whakawehea te \frac{14}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{7}{3}. Nā, tāpiria te pūrua o te \frac{7}{3} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{14}{3}x+\left(\frac{7}{3}\right)^{2}=-\frac{8}{3}+\left(\frac{7}{3}\right)^{2}$$

Pūruatia \frac{7}{3} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=-\frac{8}{3}+\frac{49}{9}$$

Tāpiri -\frac{8}{3} ki te \frac{49}{9} 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.

$$x^{2}+\frac{14}{3}x+\frac{49}{9}=\frac{25}{9}$$

Tauwehea x^{2}+\frac{14}{3}x+\frac{49}{9}. 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}.

$$\left(x+\frac{7}{3}\right)^{2}=\frac{25}{9}$$

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

$$\sqrt{\left(x+\frac{7}{3}\right)^{2}}=\sqrt{\frac{25}{9}}$$

Whakarūnātia.

$$x+\frac{7}{3}=\frac{5}{3} x+\frac{7}{3}=-\frac{5}{3}$$

Me tango \frac{7}{3} mai i ngā taha e rua o te whārite.

$$x=-\frac{2}{3} x=-4$$