Whakaoti i te 20x^2+38x+12

$$20 { x }^{ 2 } +38x+12$$

Tauwehe

$$2\left(2x+3\right)\left(5x+2\right)$$

Ngā Upane Otinga

Tauwehea te 2.

$$2\left(10x^{2}+19x+6\right)$$

Whakaarohia te 10x^{2}+19x+6. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 10x^{2}+ax+bx+6. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=19 ab=10\times 6=60$$

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 60.

$$1,60 2,30 3,20 4,15 5,12 6,10$$

Tātaihia te tapeke mō ia takirua.

$$1+60=61 2+30=32 3+20=23 4+15=19 5+12=17 6+10=16$$

Ko te otinga te takirua ka hoatu i te tapeke 19.

$$a=4 b=15$$

Tuhia anō te 10x^{2}+19x+6 hei \left(10x^{2}+4x\right)+\left(15x+6\right).

$$\left(10x^{2}+4x\right)+\left(15x+6\right)$$

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

$$2x\left(5x+2\right)+3\left(5x+2\right)$$

Whakatauwehea atu te kīanga pātahi 5x+2 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x+2\right)\left(2x+3\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$2\left(5x+2\right)\left(2x+3\right)$$

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.

$$20x^{2}+38x+12=0$$

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.

$$x=\frac{-38±\sqrt{38^{2}-4\times 20\times 12}}{2\times 20}$$

Pūrua 38.

$$x=\frac{-38±\sqrt{1444-4\times 20\times 12}}{2\times 20}$$

Whakareatia -4 ki te 20.

$$x=\frac{-38±\sqrt{1444-80\times 12}}{2\times 20}$$

Whakareatia -80 ki te 12.

$$x=\frac{-38±\sqrt{1444-960}}{2\times 20}$$

Tāpiri 1444 ki te -960.

$$x=\frac{-38±\sqrt{484}}{2\times 20}$$

Tuhia te pūtakerua o te 484.

$$x=\frac{-38±22}{2\times 20}$$

Whakareatia 2 ki te 20.

$$x=\frac{-38±22}{40}$$

Nā, me whakaoti te whārite x=\frac{-38±22}{40} ina he tāpiri te ±. Tāpiri -38 ki te 22.

$$x=-\frac{16}{40}$$

Whakahekea te hautanga \frac{-16}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Nā, me whakaoti te whārite x=\frac{-38±22}{40} ina he tango te ±. Tango 22 mai i -38.

$$x=-\frac{60}{40}$$

Whakahekea te hautanga \frac{-60}{40} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

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

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

$$20x^{2}+38x+12=20\left(x-\left(-\frac{2}{5}\right)\right)\left(x-\left(-\frac{3}{2}\right)\right)$$

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

$$20x^{2}+38x+12=20\left(x+\frac{2}{5}\right)\left(x+\frac{3}{2}\right)$$

Tāpiri \frac{2}{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.

$$20x^{2}+38x+12=20\times \frac{5x+2}{5}\left(x+\frac{3}{2}\right)$$

Tāpiri \frac{3}{2} 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.

$$20x^{2}+38x+12=20\times \frac{5x+2}{5}\times \frac{2x+3}{2}$$

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

$$20x^{2}+38x+12=20\times \frac{\left(5x+2\right)\left(2x+3\right)}{5\times 2}$$

Whakareatia 5 ki te 2.

$$20x^{2}+38x+12=20\times \frac{\left(5x+2\right)\left(2x+3\right)}{10}$$

Whakakorea atu te tauwehe pūnoa nui rawa 10 i roto i te 20 me te 10.

$$20x^{2}+38x+12=2\left(5x+2\right)\left(2x+3\right)$$

Aromātai

$$20x^{2}+38x+12$$