Whakaoti i te g(p)=-60p^2+50p+10

$$g ( p ) = - 60 p ^ { 2 } + 50 p + 10$$

Tauwehe

$$10\left(1-p\right)\left(6p+1\right)$$

Ngā Upane Otinga

Tauwehea te 10.

$$10\left(-6p^{2}+5p+1\right)$$

Whakaarohia te -6p^{2}+5p+1. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -6p^{2}+ap+bp+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=5 ab=-6=-6$$

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 -6.

$$-1,6 -2,3$$

Tātaihia te tapeke mō ia takirua.

$$-1+6=5 -2+3=1$$

Ko te otinga te takirua ka hoatu i te tapeke 5.

$$a=6 b=-1$$

Tuhia anō te -6p^{2}+5p+1 hei \left(-6p^{2}+6p\right)+\left(-p+1\right).

$$\left(-6p^{2}+6p\right)+\left(-p+1\right)$$

Whakatauwehea atu 6p i te -6p^{2}+6p.

$$6p\left(-p+1\right)-p+1$$

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

$$\left(-p+1\right)\left(6p+1\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$10\left(-p+1\right)\left(6p+1\right)$$

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.

$$-60p^{2}+50p+10=0$$

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.

$$p=\frac{-50±\sqrt{50^{2}-4\left(-60\right)\times 10}}{2\left(-60\right)}$$

Pūrua 50.

$$p=\frac{-50±\sqrt{2500-4\left(-60\right)\times 10}}{2\left(-60\right)}$$

Whakareatia -4 ki te -60.

$$p=\frac{-50±\sqrt{2500+240\times 10}}{2\left(-60\right)}$$

Whakareatia 240 ki te 10.

$$p=\frac{-50±\sqrt{2500+2400}}{2\left(-60\right)}$$

Tāpiri 2500 ki te 2400.

$$p=\frac{-50±\sqrt{4900}}{2\left(-60\right)}$$

Tuhia te pūtakerua o te 4900.

$$p=\frac{-50±70}{2\left(-60\right)}$$

Whakareatia 2 ki te -60.

$$p=\frac{-50±70}{-120}$$

Nā, me whakaoti te whārite p=\frac{-50±70}{-120} ina he tāpiri te ±. Tāpiri -50 ki te 70.

$$p=\frac{20}{-120}$$

Whakahekea te hautanga \frac{20}{-120} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

$$p=-\frac{1}{6}$$

Nā, me whakaoti te whārite p=\frac{-50±70}{-120} ina he tango te ±. Tango 70 mai i -50.

$$p=-\frac{120}{-120}$$

Whakawehe -120 ki te -120.

$$p=1$$

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}{6} mō te x_{1} me te 1 mō te x_{2}.

$$-60p^{2}+50p+10=-60\left(p-\left(-\frac{1}{6}\right)\right)\left(p-1\right)$$

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

$$-60p^{2}+50p+10=-60\left(p+\frac{1}{6}\right)\left(p-1\right)$$

Tāpiri \frac{1}{6} 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.

$$-60p^{2}+50p+10=-60\times \frac{-6p-1}{-6}\left(p-1\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 6 i roto i te -60 me te 6.

$$-60p^{2}+50p+10=10\left(-6p-1\right)\left(p-1\right)$$

Aromātai

$$10+50p-60p^{2}$$