Whakaoti i te 6x^2-x-40

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

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga 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.

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

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka taea te huamaha pūrua te tauwehe mā te whakamahi i te huringa ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), ina ko x_{1} me x_{2} ngā otinga o te whārite pūrua ax^{2}+bx+c=0.

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

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.

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

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.

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

Tauwehea te kīanga taketake mā te whakamahi i te ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Me whakakapi te \frac{8}{3} mō te x_{1} me te -\frac{5}{2} mō te x_{2}.

$$6x^{2}-x-40=6\left(x-\frac{8}{3}\right)\left(x-\left(-\frac{5}{2}\right)\right)$$

Whakamāmātia ngā kīanga katoa o te āhua p-\left(-q\right) ki te p+q.

$$6x^{2}-x-40=6\left(x-\frac{8}{3}\right)\left(x+\frac{5}{2}\right)$$

Tango \frac{8}{3} mai i x mā te kimi i te tauraro pātahi me te tango i ngā taurunga, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$6x^{2}-x-40=6\times \frac{3x-8}{3}\left(x+\frac{5}{2}\right)$$

Tāpiri \frac{5}{2} ki te x 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.

$$6x^{2}-x-40=6\times \frac{3x-8}{3}\times \frac{2x+5}{2}$$

Whakareatia \frac{3x-8}{3} ki te \frac{2x+5}{2} mā te whakarea taurunga ki te taurunga me te tauraro ki te tauraro, ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$6x^{2}-x-40=6\times \frac{\left(3x-8\right)\left(2x+5\right)}{3\times 2}$$

Whakareatia 3 ki te 2.

$$6x^{2}-x-40=6\times \frac{\left(3x-8\right)\left(2x+5\right)}{6}$$

Whakakorea atu te tauwehe pūnoa nui rawa 6 i roto i te 6 me te 6.

$$6x^{2}-x-40=\left(3x-8\right)\left(2x+5\right)$$

Aromātai

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