Whakaoti i te 2x^2+7x+5

$$2 x ^ { 2 } + 7 x + 5$$

Tauwehe

$$\left(x+1\right)\left(2x+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 2x^{2}+ax+bx+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=2\times 5=10$$

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 10.

$$1,10 2,5$$

Tātaihia te tapeke mō ia takirua.

$$1+10=11 2+5=7$$

Ko te otinga te takirua ka hoatu i te tapeke 7.

$$a=2 b=5$$

Tuhia anō te 2x^{2}+7x+5 hei \left(2x^{2}+2x\right)+\left(5x+5\right).

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

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

$$2x\left(x+1\right)+5\left(x+1\right)$$

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

$$\left(x+1\right)\left(2x+5\right)$$

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.

$$2x^{2}+7x+5=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 2\times 5}}{2\times 2}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 2\times 5}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-7±\sqrt{49-8\times 5}}{2\times 2}$$

Whakareatia -8 ki te 5.

$$x=\frac{-7±\sqrt{49-40}}{2\times 2}$$

Tāpiri 49 ki te -40.

$$x=\frac{-7±\sqrt{9}}{2\times 2}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-7±3}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{-7±3}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±3}{4} ina he tāpiri te ±. Tāpiri -7 ki te 3.

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

Whakawehe -4 ki te 4.

$$x=-1$$

Nā, me whakaoti te whārite x=\frac{-7±3}{4} ina he tango te ±. Tango 3 mai i -7.

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

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

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

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

$$2x^{2}+7x+5=2\left(x-\left(-1\right)\right)\left(x-\left(-\frac{5}{2}\right)\right)$$

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

$$2x^{2}+7x+5=2\left(x+1\right)\left(x+\frac{5}{2}\right)$$

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.

$$2x^{2}+7x+5=2\left(x+1\right)\times \frac{2x+5}{2}$$

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

$$2x^{2}+7x+5=\left(x+1\right)\left(2x+5\right)$$

Aromātai

$$\left(x+1\right)\left(2x+5\right)$$