Whakaoti i te 6v^2+17v+5

$$6 v ^ { 2 } + 17 v + 5$$

Tauwehe

$$\left(2v+5\right)\left(3v+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 6v^{2}+av+bv+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=17 ab=6\times 5=30$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. 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.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

Ko te otinga te takirua ka hoatu i te tapeke 17.

$$a=2 b=15$$

Tuhia anō te 6v^{2}+17v+5 hei \left(6v^{2}+2v\right)+\left(15v+5\right).

$$\left(6v^{2}+2v\right)+\left(15v+5\right)$$

Tauwehea te 2v i te tuatahi me te 5 i te rōpū tuarua.

$$2v\left(3v+1\right)+5\left(3v+1\right)$$

Whakatauwehea atu te kīanga pātahi 3v+1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3v+1\right)\left(2v+5\right)$$

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.

$$6v^{2}+17v+5=0$$

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.

$$v=\frac{-17±\sqrt{17^{2}-4\times 6\times 5}}{2\times 6}$$

Pūrua 17.

$$v=\frac{-17±\sqrt{289-4\times 6\times 5}}{2\times 6}$$

Whakareatia -4 ki te 6.

$$v=\frac{-17±\sqrt{289-24\times 5}}{2\times 6}$$

Whakareatia -24 ki te 5.

$$v=\frac{-17±\sqrt{289-120}}{2\times 6}$$

Tāpiri 289 ki te -120.

$$v=\frac{-17±\sqrt{169}}{2\times 6}$$

Tuhia te pūtakerua o te 169.

$$v=\frac{-17±13}{2\times 6}$$

Whakareatia 2 ki te 6.

$$v=\frac{-17±13}{12}$$

Nā, me whakaoti te whārite v=\frac{-17±13}{12} ina he tāpiri te ±. Tāpiri -17 ki te 13.

$$v=-\frac{4}{12}$$

Whakahekea te hautanga \frac{-4}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$v=-\frac{1}{3}$$

Nā, me whakaoti te whārite v=\frac{-17±13}{12} ina he tango te ±. Tango 13 mai i -17.

$$v=-\frac{30}{12}$$

Whakahekea te hautanga \frac{-30}{12} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$v=-\frac{5}{2}$$

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

$$6v^{2}+17v+5=6\left(v-\left(-\frac{1}{3}\right)\right)\left(v-\left(-\frac{5}{2}\right)\right)$$

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

$$6v^{2}+17v+5=6\left(v+\frac{1}{3}\right)\left(v+\frac{5}{2}\right)$$

Tāpiri \frac{1}{3} ki te v 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.

$$6v^{2}+17v+5=6\times \frac{3v+1}{3}\left(v+\frac{5}{2}\right)$$

Tāpiri \frac{5}{2} ki te v 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.

$$6v^{2}+17v+5=6\times \frac{3v+1}{3}\times \frac{2v+5}{2}$$

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

$$6v^{2}+17v+5=6\times \frac{\left(3v+1\right)\left(2v+5\right)}{3\times 2}$$

Whakareatia 3 ki te 2.

$$6v^{2}+17v+5=6\times \frac{\left(3v+1\right)\left(2v+5\right)}{6}$$

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

$$6v^{2}+17v+5=\left(3v+1\right)\left(2v+5\right)$$

Aromātai

$$\left(2v+5\right)\left(3v+1\right)$$