Whakaoti i te 4x^2-12x-7

$$4 { x }^{ 2 } -12x-7$$

Tauwehe

$$\left(2x-7\right)\left(2x+1\right)$$

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.

$$a+b=-12 ab=4\left(-7\right)=-28$$

I te mea kua tōraro te ab, he tauaro ngā tohu o a me b. I te mea kua tōraro te a+b, he nui ake te uara pū o te tau tōraro i tō te tōrunga. 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.

$$1-28=-27 2-14=-12 4-7=-3$$

Ko te otinga te takirua ka hoatu i te tapeke -12.

$$a=-14 b=2$$

Tuhia anō te 4x^{2}-12x-7 hei \left(4x^{2}-14x\right)+\left(2x-7\right).

$$\left(4x^{2}-14x\right)+\left(2x-7\right)$$

Whakatauwehea atu 2x i te 4x^{2}-14x.

$$2x\left(2x-7\right)+2x-7$$

Whakatauwehea atu te kīanga pātahi 2x-7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-7\right)\left(2x+1\right)$$

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.

$$4x^{2}-12x-7=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 4\left(-7\right)}}{2\times 4}$$

Pūrua -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\times 4\left(-7\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$x=\frac{-\left(-12\right)±\sqrt{144-16\left(-7\right)}}{2\times 4}$$

Whakareatia -16 ki te -7.

$$x=\frac{-\left(-12\right)±\sqrt{144+112}}{2\times 4}$$

Tāpiri 144 ki te 112.

$$x=\frac{-\left(-12\right)±\sqrt{256}}{2\times 4}$$

Tuhia te pūtakerua o te 256.

$$x=\frac{-\left(-12\right)±16}{2\times 4}$$

Ko te tauaro o -12 ko 12.

$$x=\frac{12±16}{2\times 4}$$

Whakareatia 2 ki te 4.

$$x=\frac{12±16}{8}$$

Nā, me whakaoti te whārite x=\frac{12±16}{8} ina he tāpiri te ±. Tāpiri 12 ki te 16.

$$x=\frac{28}{8}$$

Whakahekea te hautanga \frac{28}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=\frac{7}{2}$$

Nā, me whakaoti te whārite x=\frac{12±16}{8} ina he tango te ±. Tango 16 mai i 12.

$$x=-\frac{4}{8}$$

Whakahekea te hautanga \frac{-4}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{1}{2}$$

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{7}{2} mō te x_{1} me te -\frac{1}{2} mō te x_{2}.

$$4x^{2}-12x-7=4\left(x-\frac{7}{2}\right)\left(x-\left(-\frac{1}{2}\right)\right)$$

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

$$4x^{2}-12x-7=4\left(x-\frac{7}{2}\right)\left(x+\frac{1}{2}\right)$$

Tango \frac{7}{2} 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.

$$4x^{2}-12x-7=4\times \frac{2x-7}{2}\left(x+\frac{1}{2}\right)$$

Tāpiri \frac{1}{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.

$$4x^{2}-12x-7=4\times \frac{2x-7}{2}\times \frac{2x+1}{2}$$

Whakareatia \frac{2x-7}{2} ki te \frac{2x+1}{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.

$$4x^{2}-12x-7=4\times \frac{\left(2x-7\right)\left(2x+1\right)}{2\times 2}$$

Whakareatia 2 ki te 2.

$$4x^{2}-12x-7=4\times \frac{\left(2x-7\right)\left(2x+1\right)}{4}$$

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

$$4x^{2}-12x-7=\left(2x-7\right)\left(2x+1\right)$$

Aromātai

$$\left(2x-7\right)\left(2x+1\right)$$