Whakaoti i te 6x^2+11x+3=0

$$6 { x }^{ 2 } +11x+3=0$$

Whakaoti mō x

$$x = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

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

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 6x^{2}+ax+bx+3. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=6\times 3=18$$

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 18.

$$1,18 2,9 3,6$$

Tātaihia te tapeke mō ia takirua.

$$1+18=19 2+9=11 3+6=9$$

Ko te otinga te takirua ka hoatu i te tapeke 11.

$$a=2 b=9$$

Tuhia anō te 6x^{2}+11x+3 hei \left(6x^{2}+2x\right)+\left(9x+3\right).

$$\left(6x^{2}+2x\right)+\left(9x+3\right)$$

Tauwehea te 2x i te tuatahi me te 3 i te rōpū tuarua.

$$2x\left(3x+1\right)+3\left(3x+1\right)$$

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

$$\left(3x+1\right)\left(2x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 3x+1=0 me te 2x+3=0.

$$x=-\frac{1}{3} x=-\frac{3}{2}$$

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.

$$6x^{2}+11x+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, 11 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-11±\sqrt{11^{2}-4\times 6\times 3}}{2\times 6}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\times 6\times 3}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-11±\sqrt{121-24\times 3}}{2\times 6}$$

Whakareatia -24 ki te 3.

$$x=\frac{-11±\sqrt{121-72}}{2\times 6}$$

Tāpiri 121 ki te -72.

$$x=\frac{-11±\sqrt{49}}{2\times 6}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-11±7}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{-11±7}{12}$$

Nā, me whakaoti te whārite x=\frac{-11±7}{12} ina he tāpiri te ±. Tāpiri -11 ki te 7.

$$x=-\frac{4}{12}$$

Whakahekea te hautanga \frac{-4}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite x=\frac{-11±7}{12} ina he tango te ±. Tango 7 mai i -11.

$$x=-\frac{18}{12}$$

Whakahekea te hautanga \frac{-18}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Kua oti te whārite te whakatau.

$$x=-\frac{1}{3} x=-\frac{3}{2}$$

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.

$$6x^{2}+11x+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$6x^{2}+11x+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$6x^{2}+11x=-3$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}+11x}{6}=-\frac{3}{6}$$

Mā te whakawehe ki te 6 ka wetekia te whakareanga ki te 6.

$$x^{2}+\frac{11}{6}x=-\frac{3}{6}$$

Whakahekea te hautanga \frac{-3}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

$$x^{2}+\frac{11}{6}x=-\frac{1}{2}$$

Whakawehea te \frac{11}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{11}{12}. Nā, tāpiria te pūrua o te \frac{11}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{11}{6}x+\left(\frac{11}{12}\right)^{2}=-\frac{1}{2}+\left(\frac{11}{12}\right)^{2}$$

Pūruatia \frac{11}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=-\frac{1}{2}+\frac{121}{144}$$

Tāpiri -\frac{1}{2} ki te \frac{121}{144} 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.

$$x^{2}+\frac{11}{6}x+\frac{121}{144}=\frac{49}{144}$$

Tauwehea x^{2}+\frac{11}{6}x+\frac{121}{144}. 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}.

$$\left(x+\frac{11}{12}\right)^{2}=\frac{49}{144}$$

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

$$\sqrt{\left(x+\frac{11}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Whakarūnātia.

$$x+\frac{11}{12}=\frac{7}{12} x+\frac{11}{12}=-\frac{7}{12}$$

Me tango \frac{11}{12} mai i ngā taha e rua o te whārite.

$$x=-\frac{1}{3} x=-\frac{3}{2}$$