Whakaoti i te 4x^2+7x-2

$$4 { x }^{ 2 } +7x-2$$

Tauwehe

$$\left(4x-1\right)\left(x+2\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=7 ab=4\left(-2\right)=-8$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōrunga te a+b, he nui ake te uara pū o te tau tōrunga i tō te tōraro. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua -8.

$$-1,8 -2,4$$

Tātaihia te tapeke mō ia takirua.

$$-1+8=7 -2+4=2$$

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

$$a=-1 b=8$$

Tuhia anō te 4x^{2}+7x-2 hei \left(4x^{2}-x\right)+\left(8x-2\right).

$$\left(4x^{2}-x\right)+\left(8x-2\right)$$

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

$$x\left(4x-1\right)+2\left(4x-1\right)$$

Whakatauwehea atu te kīanga pātahi 4x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-1\right)\left(x+2\right)$$

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.

$$4x^{2}+7x-2=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\times 4\left(-2\right)}}{2\times 4}$$

Pūrua 7.

$$x=\frac{-7±\sqrt{49-4\times 4\left(-2\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-7±\sqrt{49-16\left(-2\right)}}{2\times 4}$$

Whakareatia -16 ki te -2.

$$x=\frac{-7±\sqrt{49+32}}{2\times 4}$$

Tāpiri 49 ki te 32.

$$x=\frac{-7±\sqrt{81}}{2\times 4}$$

Tuhia te pūtakerua o te 81.

$$x=\frac{-7±9}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{-7±9}{8}$$

Nā, me whakaoti te whārite x=\frac{-7±9}{8} ina he tāpiri te ±. Tāpiri -7 ki te 9.

$$x=\frac{2}{8}$$

Whakahekea te hautanga \frac{2}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{4}$$

Nā, me whakaoti te whārite x=\frac{-7±9}{8} ina he tango te ±. Tango 9 mai i -7.

$$x=-\frac{16}{8}$$

Whakawehe -16 ki te 8.

$$x=-2$$

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}{4} mō te x_{1} me te -2 mō te x_{2}.

$$4x^{2}+7x-2=4\left(x-\frac{1}{4}\right)\left(x-\left(-2\right)\right)$$

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

$$4x^{2}+7x-2=4\left(x-\frac{1}{4}\right)\left(x+2\right)$$

Tango \frac{1}{4} mai i x 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.

$$4x^{2}+7x-2=4\times \frac{4x-1}{4}\left(x+2\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te 4 me te 4.

$$4x^{2}+7x-2=\left(4x-1\right)\left(x+2\right)$$

Aromātai

$$\left(4x-1\right)\left(x+2\right)$$