Whakaoti i te 2x^2+11x+15

$$2 x ^ { 2 } + 11 x + 15$$

Tauwehe

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

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2x^{2}+ax+bx+15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=11 ab=2\times 15=30$$

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 11.

$$a=5 b=6$$

Tuhia anō te 2x^{2}+11x+15 hei \left(2x^{2}+5x\right)+\left(6x+15\right).

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

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

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

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

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

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.

$$2x^{2}+11x+15=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 2\times 15}}{2\times 2}$$

Pūrua 11.

$$x=\frac{-11±\sqrt{121-4\times 2\times 15}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-11±\sqrt{121-8\times 15}}{2\times 2}$$

Whakareatia -8 ki te 15.

$$x=\frac{-11±\sqrt{121-120}}{2\times 2}$$

Tāpiri 121 ki te -120.

$$x=\frac{-11±\sqrt{1}}{2\times 2}$$

Tuhia te pūtakerua o te 1.

$$x=\frac{-11±1}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-11±1}{4}$$

Nā, me whakaoti te whārite x=\frac{-11±1}{4} ina he tāpiri te ±. Tāpiri -11 ki te 1.

$$x=-\frac{10}{4}$$

Whakahekea te hautanga \frac{-10}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Nā, me whakaoti te whārite x=\frac{-11±1}{4} ina he tango te ±. Tango 1 mai i -11.

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

Whakawehe -12 ki te 4.

$$x=-3$$

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

$$2x^{2}+11x+15=2\left(x-\left(-\frac{5}{2}\right)\right)\left(x-\left(-3\right)\right)$$

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

$$2x^{2}+11x+15=2\left(x+\frac{5}{2}\right)\left(x+3\right)$$

Tāpiri \frac{5}{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.

$$2x^{2}+11x+15=2\times \frac{2x+5}{2}\left(x+3\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 2 i roto i te 2 me te 2.

$$2x^{2}+11x+15=\left(2x+5\right)\left(x+3\right)$$

Aromātai

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