Whakaoti i te 6x^2-x=40

$$6 x ^ { 2 } - x = 40$$

Whakaoti mō x

$$x = -\frac{5}{2} = -2\frac{1}{2} = -2.5$$

$$x = \frac{8}{3} = 2\frac{2}{3} \approx 2.666666667$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 40 mai i ngā taha e rua.

$$6x^{2}-x-40=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 6x^{2}+ax+bx-40. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-1 ab=6\left(-40\right)=-240$$

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 -240.

$$1,-240 2,-120 3,-80 4,-60 5,-48 6,-40 8,-30 10,-24 12,-20 15,-16$$

Tātaihia te tapeke mō ia takirua.

$$1-240=-239 2-120=-118 3-80=-77 4-60=-56 5-48=-43 6-40=-34 8-30=-22 10-24=-14 12-20=-8 15-16=-1$$

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

$$a=-16 b=15$$

Tuhia anō te 6x^{2}-x-40 hei \left(6x^{2}-16x\right)+\left(15x-40\right).

$$\left(6x^{2}-16x\right)+\left(15x-40\right)$$

Tauwehea te 2x i te tuatahi me te 5 i te rōpū tuarua.

$$2x\left(3x-8\right)+5\left(3x-8\right)$$

Whakatauwehea atu te kīanga pātahi 3x-8 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-8\right)\left(2x+5\right)$$

Hei kimi otinga whārite, me whakaoti te 3x-8=0 me te 2x+5=0.

$$x=\frac{8}{3} x=-\frac{5}{2}$$

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.

$$6x^{2}-x=40$$

Me tango 40 mai i ngā taha e rua o te whārite.

$$6x^{2}-x-40=40-40$$

Mā te tango i te 40 i a ia ake anō ka toe ko te 0.

$$6x^{2}-x-40=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 6 mō a, -1 mō b, me -40 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-1\right)±\sqrt{1-4\times 6\left(-40\right)}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$x=\frac{-\left(-1\right)±\sqrt{1-24\left(-40\right)}}{2\times 6}$$

Whakareatia -24 ki te -40.

$$x=\frac{-\left(-1\right)±\sqrt{1+960}}{2\times 6}$$

Tāpiri 1 ki te 960.

$$x=\frac{-\left(-1\right)±\sqrt{961}}{2\times 6}$$

Tuhia te pūtakerua o te 961.

$$x=\frac{-\left(-1\right)±31}{2\times 6}$$

Ko te tauaro o -1 ko 1.

$$x=\frac{1±31}{2\times 6}$$

Whakareatia 2 ki te 6.

$$x=\frac{1±31}{12}$$

Nā, me whakaoti te whārite x=\frac{1±31}{12} ina he tāpiri te ±. Tāpiri 1 ki te 31.

$$x=\frac{32}{12}$$

Whakahekea te hautanga \frac{32}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{8}{3}$$

Nā, me whakaoti te whārite x=\frac{1±31}{12} ina he tango te ±. Tango 31 mai i 1.

$$x=-\frac{30}{12}$$

Whakahekea te hautanga \frac{-30}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Kua oti te whārite te whakatau.

$$x=\frac{8}{3} x=-\frac{5}{2}$$

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.

$$6x^{2}-x=40$$

Whakawehea ngā taha e rua ki te 6.

$$\frac{6x^{2}-x}{6}=\frac{40}{6}$$

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

$$x^{2}-\frac{1}{6}x=\frac{40}{6}$$

Whakahekea te hautanga \frac{40}{6} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{1}{6}x=\frac{20}{3}$$

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.

$$x^{2}-\frac{1}{6}x+\left(-\frac{1}{12}\right)^{2}=\frac{20}{3}+\left(-\frac{1}{12}\right)^{2}$$

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

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{20}{3}+\frac{1}{144}$$

Tāpiri \frac{20}{3} 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.

$$x^{2}-\frac{1}{6}x+\frac{1}{144}=\frac{961}{144}$$

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}.

$$\left(x-\frac{1}{12}\right)^{2}=\frac{961}{144}$$

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

$$\sqrt{\left(x-\frac{1}{12}\right)^{2}}=\sqrt{\frac{961}{144}}$$

Whakarūnātia.

$$x-\frac{1}{12}=\frac{31}{12} x-\frac{1}{12}=-\frac{31}{12}$$

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

$$x=\frac{8}{3} x=-\frac{5}{2}$$