Whakaoti i te 6x^2-x=15

$$6 x ^ { 2 } - x = 15$$

Whakaoti mō x

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

$$x = \frac{5}{3} = 1\frac{2}{3} \approx 1.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 15 mai i ngā taha e rua.

$$6x^{2}-x-15=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 6x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=6\left(-15\right)=-90$$

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 -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -1.

$$a=-10 b=9$$

Tuhia anō te 6x^{2}-x-15 hei \left(6x^{2}-10x\right)+\left(9x-15\right).

$$\left(6x^{2}-10x\right)+\left(9x-15\right)$$

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

$$2x\left(3x-5\right)+3\left(3x-5\right)$$

Whakatauwehea atu te kīanga pātahi 3x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-5\right)\left(2x+3\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-5=0 me te 2x+3=0.

$$x=\frac{5}{3} x=-\frac{3}{2}$$

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.

$$6x^{2}-x=15$$

Me tango 15 mai i ngā taha e rua o te whārite.

$$6x^{2}-x-15=15-15$$

Mā te tango i te 15 i a ia ake anō ka toe ko te 0.

$$6x^{2}-x-15=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -1 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 6\left(-15\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-1\right)±\sqrt{1-24\left(-15\right)}}{2\times 6}$$

Whakareatia -24 ki te -15.

$$x=\frac{-\left(-1\right)±\sqrt{1+360}}{2\times 6}$$

Tāpiri 1 ki te 360.

$$x=\frac{-\left(-1\right)±\sqrt{361}}{2\times 6}$$

Tuhia te pūtakerua o te 361.

$$x=\frac{-\left(-1\right)±19}{2\times 6}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±19}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{1±19}{12}$$

Nā, me whakaoti te whārite x=\frac{1±19}{12} ina he tāpiri te ±. Tāpiri 1 ki te 19.

$$x=\frac{20}{12}$$

Whakahekea te hautanga \frac{20}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{5}{3}$$

Nā, me whakaoti te whārite x=\frac{1±19}{12} ina he tango te ±. Tango 19 mai i 1.

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

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.

$$x=\frac{5}{3} x=-\frac{3}{2}$$

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.

$$6x^{2}-x=15$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-x}{6}=\frac{15}{6}$$

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

$$x^{2}-\frac{1}{6}x=\frac{15}{6}$$

Whakahekea te hautanga \frac{15}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 3.

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

Whakawehea te -\frac{1}{6}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{12}. Nā, tāpiria te pūrua o te -\frac{1}{12} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{12} mā te pūrua i te taurunga me te tauraro o te hautanga.

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

Tāpiri \frac{5}{2} ki te \frac{1}{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.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{361}{144}$$

Tauwehea x^{2}-\frac{1}{6}x+\frac{1}{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}.

$$\left(x-\frac{1}{12}\right)^{2}=\frac{361}{144}$$

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

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{\frac{361}{144}}$$

Whakarūnātia.

$$x-\frac{1}{12}=\frac{19}{12} x-\frac{1}{12}=-\frac{19}{12}$$

Me tāpiri \frac{1}{12} ki ngā taha e rua o te whārite.

$$x=\frac{5}{3} x=-\frac{3}{2}$$