Whakaoti i te 6x^2-5x-1=0

$$6 { x }^{ 2 } -5x-1=0$$

Whakaoti mō x

$$x=-\frac{1}{6}\approx -0.166666667$$

$$x=1$$

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-1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=6\left(-1\right)=-6$$

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 -6.

$$1,-6 2,-3$$

Tātaihia te tapeke mō ia takirua.

$$1-6=-5 2-3=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -5.

$$a=-6 b=1$$

Tuhia anō te 6x^{2}-5x-1 hei \left(6x^{2}-6x\right)+\left(x-1\right).

$$\left(6x^{2}-6x\right)+\left(x-1\right)$$

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

$$6x\left(x-1\right)+x-1$$

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

$$\left(x-1\right)\left(6x+1\right)$$

Hei kimi otinga whārite, me whakaoti te x-1=0 me te 6x+1=0.

$$x=1 x=-\frac{1}{6}$$

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.

$$6x^{2}-5x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -5 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-5\right)±\sqrt{\left(-5\right)^{2}-4\times 6\left(-1\right)}}{2\times 6}$$

Pūrua -5.

$$x=\frac{-\left(-5\right)±\sqrt{25-4\times 6\left(-1\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-5\right)±\sqrt{25-24\left(-1\right)}}{2\times 6}$$

Whakareatia -24 ki te -1.

$$x=\frac{-\left(-5\right)±\sqrt{25+24}}{2\times 6}$$

Tāpiri 25 ki te 24.

$$x=\frac{-\left(-5\right)±\sqrt{49}}{2\times 6}$$

Tuhia te pūtakerua o te 49.

$$x=\frac{-\left(-5\right)±7}{2\times 6}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±7}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{5±7}{12}$$

Nā, me whakaoti te whārite x=\frac{5±7}{12} ina he tāpiri te ±. Tāpiri 5 ki te 7.

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

Whakawehe 12 ki te 12.

$$x=1$$

Nā, me whakaoti te whārite x=\frac{5±7}{12} ina he tango te ±. Tango 7 mai i 5.

$$x=-\frac{2}{12}$$

Whakahekea te hautanga \frac{-2}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{1}{6}$$

Kua oti te whārite te whakatau.

$$x=1 x=-\frac{1}{6}$$

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.

$$6x^{2}-5x-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$6x^{2}-5x-1-\left(-1\right)=-\left(-1\right)$$

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

$$6x^{2}-5x=-\left(-1\right)$$

Tango -1 mai i 0.

$$6x^{2}-5x=1$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-5x}{6}=\frac{1}{6}$$

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

$$x^{2}-\frac{5}{6}x=\frac{1}{6}$$

Whakawehea te -\frac{5}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{12}. Nā, tāpiria te pūrua o te -\frac{5}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{6}x+\left(-\frac{5}{12}\right)^{2}=\frac{1}{6}+\left(-\frac{5}{12}\right)^{2}$$

Pūruatia -\frac{5}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{1}{6}+\frac{25}{144}$$

Tāpiri \frac{1}{6} ki te \frac{25}{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.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{49}{144}$$

Tauwehea x^{2}-\frac{5}{6}x+\frac{25}{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}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{49}{144}$$

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

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{49}{144}}$$

Whakarūnātia.

$$x-\frac{5}{12}=\frac{7}{12} x-\frac{5}{12}=-\frac{7}{12}$$

Me tāpiri \frac{5}{12} ki ngā taha e rua o te whārite.

$$x=1 x=-\frac{1}{6}$$