Whakaoti i te 10a^2-13a-3

$$10 a ^ { 2 } - 13 a - 3$$

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 10a^{2}+pa+qa-3. Hei kimi p me q, whakaritea tētahi pūnaha kia whakaoti.

$$p+q=-13 pq=10\left(-3\right)=-30$$

I te mea kua tōraro te pq, he tauaro ngā tohu o p me q. I te mea kua tōraro te p+q, 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 -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 -13.

$$p=-15 q=2$$

Tuhia anō te 10a^{2}-13a-3 hei \left(10a^{2}-15a\right)+\left(2a-3\right).

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

Whakatauwehea atu 5a i te 10a^{2}-15a.

$$5a\left(2a-3\right)+2a-3$$

Whakatauwehea atu te kīanga pātahi 2a-3 mā te whakamahi i te āhuatanga tātai tohatoha.

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

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.

$$10a^{2}-13a-3=0$$

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.

$$a=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 10\left(-3\right)}}{2\times 10}$$

Pūrua -13.

$$a=\frac{-\left(-13\right)±\sqrt{169-4\times 10\left(-3\right)}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$a=\frac{-\left(-13\right)±\sqrt{169-40\left(-3\right)}}{2\times 10}$$

Whakareatia -40 ki te -3.

$$a=\frac{-\left(-13\right)±\sqrt{169+120}}{2\times 10}$$

Tāpiri 169 ki te 120.

$$a=\frac{-\left(-13\right)±\sqrt{289}}{2\times 10}$$

Tuhia te pūtakerua o te 289.

$$a=\frac{-\left(-13\right)±17}{2\times 10}$$

Ko te tauaro o -13 ko 13.

$$a=\frac{13±17}{2\times 10}$$

Whakareatia 2 ki te 10.

$$a=\frac{13±17}{20}$$

Nā, me whakaoti te whārite a=\frac{13±17}{20} ina he tāpiri te ±. Tāpiri 13 ki te 17.

$$a=\frac{30}{20}$$

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

$$a=\frac{3}{2}$$

Nā, me whakaoti te whārite a=\frac{13±17}{20} ina he tango te ±. Tango 17 mai i 13.

$$a=-\frac{4}{20}$$

Whakahekea te hautanga \frac{-4}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$a=-\frac{1}{5}$$

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

$$10a^{2}-13a-3=10\left(a-\frac{3}{2}\right)\left(a-\left(-\frac{1}{5}\right)\right)$$

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

$$10a^{2}-13a-3=10\left(a-\frac{3}{2}\right)\left(a+\frac{1}{5}\right)$$

Tango \frac{3}{2} mai i a 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.

$$10a^{2}-13a-3=10\times \frac{2a-3}{2}\left(a+\frac{1}{5}\right)$$

Tāpiri \frac{1}{5} ki te a 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.

$$10a^{2}-13a-3=10\times \frac{2a-3}{2}\times \frac{5a+1}{5}$$

Whakareatia \frac{2a-3}{2} ki te \frac{5a+1}{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.

$$10a^{2}-13a-3=10\times \frac{\left(2a-3\right)\left(5a+1\right)}{2\times 5}$$

Whakareatia 2 ki te 5.

$$10a^{2}-13a-3=10\times \frac{\left(2a-3\right)\left(5a+1\right)}{10}$$

Whakakorea atu te tauwehe pūnoa nui rawa 10 i roto i te 10 me te 10.

$$10a^{2}-13a-3=\left(2a-3\right)\left(5a+1\right)$$

Aromātai

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