Whakaoti i te 4x^2-11x+7

$$4 { x }^{ 2 } -11x+7$$

Tauwehe

$$\left(x-1\right)\left(4x-7\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4x^{2}+ax+bx+7. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=4\times 7=28$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 28.

$$-1,-28 -2,-14 -4,-7$$

Tātaihia te tapeke mō ia takirua.

$$-1-28=-29 -2-14=-16 -4-7=-11$$

Ko te otinga te takirua ka hoatu i te tapeke -11.

$$a=-7 b=-4$$

Tuhia anō te 4x^{2}-11x+7 hei \left(4x^{2}-7x\right)+\left(-4x+7\right).

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

Tauwehea te x i te tuatahi me te -1 i te rōpū tuarua.

$$x\left(4x-7\right)-\left(4x-7\right)$$

Whakatauwehea atu te kīanga pātahi 4x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(4x-7\right)\left(x-1\right)$$

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.

$$4x^{2}-11x+7=0$$

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.

$$x=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 4\times 7}}{2\times 4}$$

Pūrua -11.

$$x=\frac{-\left(-11\right)±\sqrt{121-4\times 4\times 7}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-11\right)±\sqrt{121-16\times 7}}{2\times 4}$$

Whakareatia -16 ki te 7.

$$x=\frac{-\left(-11\right)±\sqrt{121-112}}{2\times 4}$$

Tāpiri 121 ki te -112.

$$x=\frac{-\left(-11\right)±\sqrt{9}}{2\times 4}$$

Tuhia te pūtakerua o te 9.

$$x=\frac{-\left(-11\right)±3}{2\times 4}$$

Ko te tauaro o -11 ko 11.

$$x=\frac{11±3}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{11±3}{8}$$

Nā, me whakaoti te whārite x=\frac{11±3}{8} ina he tāpiri te ±. Tāpiri 11 ki te 3.

$$x=\frac{14}{8}$$

Whakahekea te hautanga \frac{14}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{7}{4}$$

Nā, me whakaoti te whārite x=\frac{11±3}{8} ina he tango te ±. Tango 3 mai i 11.

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

Whakawehe 8 ki te 8.

$$x=1$$

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{7}{4} mō te x_{1} me te 1 mō te x_{2}.

$$4x^{2}-11x+7=4\left(x-\frac{7}{4}\right)\left(x-1\right)$$

Tango \frac{7}{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.

$$4x^{2}-11x+7=4\times \frac{4x-7}{4}\left(x-1\right)$$

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

$$4x^{2}-11x+7=\left(4x-7\right)\left(x-1\right)$$

Aromātai

$$\left(x-1\right)\left(4x-7\right)$$