Whakaoti i te 6x^2-7x-3=0

$$6 x ^ { 2 } - 7 x - 3 = 0$$

Whakaoti mō x

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

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

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.

$$a+b=-7 ab=6\left(-3\right)=-18$$

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 -18.

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

Tātaihia te tapeke mō ia takirua.

$$1-18=-17 2-9=-7 3-6=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -7.

$$a=-9 b=2$$

Tuhia anō te 6x^{2}-7x-3 hei \left(6x^{2}-9x\right)+\left(2x-3\right).

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

Whakatauwehea atu 3x i te 6x^{2}-9x.

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

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

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

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

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

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.

$$6x^{2}-7x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -7 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 6\left(-3\right)}}{2\times 6}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 6\left(-3\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-7\right)±\sqrt{49-24\left(-3\right)}}{2\times 6}$$

Whakareatia -24 ki te -3.

$$x=\frac{-\left(-7\right)±\sqrt{49+72}}{2\times 6}$$

Tāpiri 49 ki te 72.

$$x=\frac{-\left(-7\right)±\sqrt{121}}{2\times 6}$$

Tuhia te pūtakerua o te 121.

$$x=\frac{-\left(-7\right)±11}{2\times 6}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±11}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{7±11}{12}$$

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

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

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

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

Kua oti te whārite te whakatau.

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

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.

$$6x^{2}-7x-3=0$$

Me tāpiri 3 ki ngā taha e rua o te whārite.

$$6x^{2}-7x-3-\left(-3\right)=-\left(-3\right)$$

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

$$6x^{2}-7x=-\left(-3\right)$$

Tango -3 mai i 0.

$$6x^{2}-7x=3$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-7x}{6}=\frac{3}{6}$$

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

$$x^{2}-\frac{7}{6}x=\frac{3}{6}$$

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

$$x^{2}-\frac{7}{6}x=\frac{1}{2}$$

Whakawehea te -\frac{7}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{12}. Nā, tāpiria te pūrua o te -\frac{7}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{6}x+\left(-\frac{7}{12}\right)^{2}=\frac{1}{2}+\left(-\frac{7}{12}\right)^{2}$$

Pūruatia -\frac{7}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=\frac{1}{2}+\frac{49}{144}$$

Tāpiri \frac{1}{2} ki te \frac{49}{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.

$$x^{2}-\frac{7}{6}x+\frac{49}{144}=\frac{121}{144}$$

Tauwehea te x^{2}-\frac{7}{6}x+\frac{49}{144}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{12}\right)^{2}=\frac{121}{144}$$

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

$$\sqrt{\left(x-\frac{7}{12}\right)^{2}}=\sqrt{\frac{121}{144}}$$

Whakarūnātia.

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

Me tāpiri \frac{7}{12} ki ngā taha e rua o te whārite.

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