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

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

Whakaoti mō x

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

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

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-6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-5 ab=6\left(-6\right)=-36$$

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 -36.

$$1,-36 2,-18 3,-12 4,-9 6,-6$$

Tātaihia te tapeke mō ia takirua.

$$1-36=-35 2-18=-16 3-12=-9 4-9=-5 6-6=0$$

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

$$a=-9 b=4$$

Tuhia anō te 6x^{2}-5x-6 hei \left(6x^{2}-9x\right)+\left(4x-6\right).

$$\left(6x^{2}-9x\right)+\left(4x-6\right)$$

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

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

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

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

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

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

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.

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

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

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

Pūrua -5.

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

Whakareatia -4 ki te 6.

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

Whakareatia -24 ki te -6.

$$x=\frac{-\left(-5\right)±\sqrt{25+144}}{2\times 6}$$

Tāpiri 25 ki te 144.

$$x=\frac{-\left(-5\right)±\sqrt{169}}{2\times 6}$$

Tuhia te pūtakerua o te 169.

$$x=\frac{-\left(-5\right)±13}{2\times 6}$$

Ko te tauaro o -5 ko 5.

$$x=\frac{5±13}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{5±13}{12}$$

Nā, me whakaoti te whārite x=\frac{5±13}{12} ina he tāpiri te ±. Tāpiri 5 ki te 13.

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

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{5±13}{12} ina he tango te ±. Tango 13 mai i 5.

$$x=-\frac{8}{12}$$

Whakahekea te hautanga \frac{-8}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Kua oti te whārite te whakatau.

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

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.

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

Me tāpiri 6 ki ngā taha e rua o te whārite.

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

Mā te tango i te -6 i a ia ake anō ka toe ko te 0.

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

Tango -6 mai i 0.

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

Whakawehea ngā taha e rua ki te 6.

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

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

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

Whakawehe 6 ki te 6.

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

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.

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

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

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

Tāpiri 1 ki te \frac{25}{144}.

$$x^{2}-\frac{5}{6}x+\frac{25}{144}=\frac{169}{144}$$

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}.

$$\left(x-\frac{5}{12}\right)^{2}=\frac{169}{144}$$

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

$$\sqrt{\left(x-\frac{5}{12}\right)^{2}}=\sqrt{\frac{169}{144}}$$

Whakarūnātia.

$$x-\frac{5}{12}=\frac{13}{12} x-\frac{5}{12}=-\frac{13}{12}$$

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

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