Whakaoti i te 80x^2+23x-15

$$80 { x }^{ 2 } +23x-15$$

Tauwehe

$$\left(16x-5\right)\left(5x+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 80x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=23 ab=80\left(-15\right)=-1200$$

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 -1200.

$$-1,1200 -2,600 -3,400 -4,300 -5,240 -6,200 -8,150 -10,120 -12,100 -15,80 -16,75 -20,60 -24,50 -25,48 -30,40$$

Tātaihia te tapeke mō ia takirua.

$$-1+1200=1199 -2+600=598 -3+400=397 -4+300=296 -5+240=235 -6+200=194 -8+150=142 -10+120=110 -12+100=88 -15+80=65 -16+75=59 -20+60=40 -24+50=26 -25+48=23 -30+40=10$$

Ko te otinga te takirua ka hoatu i te tapeke 23.

$$a=-25 b=48$$

Tuhia anō te 80x^{2}+23x-15 hei \left(80x^{2}-25x\right)+\left(48x-15\right).

$$\left(80x^{2}-25x\right)+\left(48x-15\right)$$

Tauwehea te 5x i te tuatahi me te 3 i te rōpū tuarua.

$$5x\left(16x-5\right)+3\left(16x-5\right)$$

Whakatauwehea atu te kīanga pātahi 16x-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(16x-5\right)\left(5x+3\right)$$

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.

$$80x^{2}+23x-15=0$$

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.

$$x=\frac{-23±\sqrt{23^{2}-4\times 80\left(-15\right)}}{2\times 80}$$

Pūrua 23.

$$x=\frac{-23±\sqrt{529-4\times 80\left(-15\right)}}{2\times 80}$$

Whakareatia -4 ki te 80.

$$x=\frac{-23±\sqrt{529-320\left(-15\right)}}{2\times 80}$$

Whakareatia -320 ki te -15.

$$x=\frac{-23±\sqrt{529+4800}}{2\times 80}$$

Tāpiri 529 ki te 4800.

$$x=\frac{-23±\sqrt{5329}}{2\times 80}$$

Tuhia te pūtakerua o te 5329.

$$x=\frac{-23±73}{2\times 80}$$

Whakareatia 2 ki te 80.

$$x=\frac{-23±73}{160}$$

Nā, me whakaoti te whārite x=\frac{-23±73}{160} ina he tāpiri te ±. Tāpiri -23 ki te 73.

$$x=\frac{50}{160}$$

Whakahekea te hautanga \frac{50}{160} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=\frac{5}{16}$$

Nā, me whakaoti te whārite x=\frac{-23±73}{160} ina he tango te ±. Tango 73 mai i -23.

$$x=-\frac{96}{160}$$

Whakahekea te hautanga \frac{-96}{160} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 32.

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

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

$$80x^{2}+23x-15=80\left(x-\frac{5}{16}\right)\left(x-\left(-\frac{3}{5}\right)\right)$$

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

$$80x^{2}+23x-15=80\left(x-\frac{5}{16}\right)\left(x+\frac{3}{5}\right)$$

Tango \frac{5}{16} 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.

$$80x^{2}+23x-15=80\times \frac{16x-5}{16}\left(x+\frac{3}{5}\right)$$

Tāpiri \frac{3}{5} 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.

$$80x^{2}+23x-15=80\times \frac{16x-5}{16}\times \frac{5x+3}{5}$$

Whakareatia \frac{16x-5}{16} ki te \frac{5x+3}{5} 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.

$$80x^{2}+23x-15=80\times \frac{\left(16x-5\right)\left(5x+3\right)}{16\times 5}$$

Whakareatia 16 ki te 5.

$$80x^{2}+23x-15=80\times \frac{\left(16x-5\right)\left(5x+3\right)}{80}$$

Whakakorea atu te tauwehe pūnoa nui rawa 80 i roto i te 80 me te 80.

$$80x^{2}+23x-15=\left(16x-5\right)\left(5x+3\right)$$

Aromātai

$$\left(16x-5\right)\left(5x+3\right)$$