Whakaoti i te 24x^2+x-10

$$24 x ^ { 2 } + x - 10$$

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 24x^{2}+ax+bx-10. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=1 ab=24\left(-10\right)=-240$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. 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.

$$a=-15 b=16$$

Tuhia anō te 24x^{2}+x-10 hei \left(24x^{2}-15x\right)+\left(16x-10\right).

$$\left(24x^{2}-15x\right)+\left(16x-10\right)$$

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

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

Whakatauwehea atu te kīanga pātahi 8x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

$$24x^{2}+x-10=0$$

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.

$$x=\frac{-1±\sqrt{1^{2}-4\times 24\left(-10\right)}}{2\times 24}$$

Pūrua 1.

$$x=\frac{-1±\sqrt{1-4\times 24\left(-10\right)}}{2\times 24}$$

Whakareatia -4 ki te 24.

$$x=\frac{-1±\sqrt{1-96\left(-10\right)}}{2\times 24}$$

Whakareatia -96 ki te -10.

$$x=\frac{-1±\sqrt{1+960}}{2\times 24}$$

Tāpiri 1 ki te 960.

$$x=\frac{-1±\sqrt{961}}{2\times 24}$$

Tuhia te pūtakerua o te 961.

$$x=\frac{-1±31}{2\times 24}$$

Whakareatia 2 ki te 24.

$$x=\frac{-1±31}{48}$$

Nā, me whakaoti te whārite x=\frac{-1±31}{48} ina he tāpiri te ±. Tāpiri -1 ki te 31.

$$x=\frac{30}{48}$$

Whakahekea te hautanga \frac{30}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{5}{8}$$

Nā, me whakaoti te whārite x=\frac{-1±31}{48} ina he tango te ±. Tango 31 mai i -1.

$$x=-\frac{32}{48}$$

Whakahekea te hautanga \frac{-32}{48} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

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

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{5}{8} mō te x_{1} me te -\frac{2}{3} mō te x_{2}.

$$24x^{2}+x-10=24\left(x-\frac{5}{8}\right)\left(x-\left(-\frac{2}{3}\right)\right)$$

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

$$24x^{2}+x-10=24\left(x-\frac{5}{8}\right)\left(x+\frac{2}{3}\right)$$

Tango \frac{5}{8} 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.

$$24x^{2}+x-10=24\times \frac{8x-5}{8}\left(x+\frac{2}{3}\right)$$

Tāpiri \frac{2}{3} 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.

$$24x^{2}+x-10=24\times \frac{8x-5}{8}\times \frac{3x+2}{3}$$

Whakareatia \frac{8x-5}{8} ki te \frac{3x+2}{3} 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.

$$24x^{2}+x-10=24\times \frac{\left(8x-5\right)\left(3x+2\right)}{8\times 3}$$

Whakareatia 8 ki te 3.

$$24x^{2}+x-10=24\times \frac{\left(8x-5\right)\left(3x+2\right)}{24}$$

Whakakorea atu te tauwehe pūnoa nui rawa 24 i roto i te 24 me te 24.

$$24x^{2}+x-10=\left(8x-5\right)\left(3x+2\right)$$

Aromātai

$$24x^{2}+x-10$$