Whakaoti i te 15p^2+7p-2

$$15 p ^ { 2 } + 7 p - 2$$

Tauwehe

$$\left(5p-1\right)\left(3p+2\right)$$

Ngā Upane Otinga

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

$$a+b=7 ab=15\left(-2\right)=-30$$

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 -30.

$$-1,30 -2,15 -3,10 -5,6$$

Tātaihia te tapeke mō ia takirua.

$$-1+30=29 -2+15=13 -3+10=7 -5+6=1$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=-3 b=10$$

Tuhia anō te 15p^{2}+7p-2 hei \left(15p^{2}-3p\right)+\left(10p-2\right).

$$\left(15p^{2}-3p\right)+\left(10p-2\right)$$

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

$$3p\left(5p-1\right)+2\left(5p-1\right)$$

Whakatauwehea atu te kīanga pātahi 5p-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5p-1\right)\left(3p+2\right)$$

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.

$$15p^{2}+7p-2=0$$

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.

$$p=\frac{-7±\sqrt{7^{2}-4\times 15\left(-2\right)}}{2\times 15}$$

Pūrua 7.

$$p=\frac{-7±\sqrt{49-4\times 15\left(-2\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$p=\frac{-7±\sqrt{49-60\left(-2\right)}}{2\times 15}$$

Whakareatia -60 ki te -2.

$$p=\frac{-7±\sqrt{49+120}}{2\times 15}$$

Tāpiri 49 ki te 120.

$$p=\frac{-7±\sqrt{169}}{2\times 15}$$

Tuhia te pūtakerua o te 169.

$$p=\frac{-7±13}{2\times 15}$$

Whakareatia 2 ki te 15.

$$p=\frac{-7±13}{30}$$

Nā, me whakaoti te whārite p=\frac{-7±13}{30} ina he tāpiri te ±. Tāpiri -7 ki te 13.

$$p=\frac{6}{30}$$

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

$$p=\frac{1}{5}$$

Nā, me whakaoti te whārite p=\frac{-7±13}{30} ina he tango te ±. Tango 13 mai i -7.

$$p=-\frac{20}{30}$$

Whakahekea te hautanga \frac{-20}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

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

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

$$15p^{2}+7p-2=15\left(p-\frac{1}{5}\right)\left(p-\left(-\frac{2}{3}\right)\right)$$

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

$$15p^{2}+7p-2=15\left(p-\frac{1}{5}\right)\left(p+\frac{2}{3}\right)$$

Tango \frac{1}{5} mai i p 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.

$$15p^{2}+7p-2=15\times \frac{5p-1}{5}\left(p+\frac{2}{3}\right)$$

Tāpiri \frac{2}{3} ki te p 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.

$$15p^{2}+7p-2=15\times \frac{5p-1}{5}\times \frac{3p+2}{3}$$

Whakareatia \frac{5p-1}{5} ki te \frac{3p+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.

$$15p^{2}+7p-2=15\times \frac{\left(5p-1\right)\left(3p+2\right)}{5\times 3}$$

Whakareatia 5 ki te 3.

$$15p^{2}+7p-2=15\times \frac{\left(5p-1\right)\left(3p+2\right)}{15}$$

Whakakorea atu te tauwehe pūnoa nui rawa 15 i roto i te 15 me te 15.

$$15p^{2}+7p-2=\left(5p-1\right)\left(3p+2\right)$$

Aromātai

$$\left(5p-1\right)\left(3p+2\right)$$