Whakaoti i te 7x^2+18x-9

$$7 x ^ { 2 } + 18 x - 9$$

Tauwehe

$$\left(7x-3\right)\left(x+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 7x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=18 ab=7\left(-9\right)=-63$$

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 -63.

$$-1,63 -3,21 -7,9$$

Tātaihia te tapeke mō ia takirua.

$$-1+63=62 -3+21=18 -7+9=2$$

Ko te otinga te takirua ka hoatu i te tapeke 18.

$$a=-3 b=21$$

Tuhia anō te 7x^{2}+18x-9 hei \left(7x^{2}-3x\right)+\left(21x-9\right).

$$\left(7x^{2}-3x\right)+\left(21x-9\right)$$

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

$$x\left(7x-3\right)+3\left(7x-3\right)$$

Whakatauwehea atu te kīanga pātahi 7x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(7x-3\right)\left(x+3\right)$$

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.

$$7x^{2}+18x-9=0$$

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.

$$x=\frac{-18±\sqrt{18^{2}-4\times 7\left(-9\right)}}{2\times 7}$$

Pūrua 18.

$$x=\frac{-18±\sqrt{324-4\times 7\left(-9\right)}}{2\times 7}$$

Whakareatia -4 ki te 7.

$$x=\frac{-18±\sqrt{324-28\left(-9\right)}}{2\times 7}$$

Whakareatia -28 ki te -9.

$$x=\frac{-18±\sqrt{324+252}}{2\times 7}$$

Tāpiri 324 ki te 252.

$$x=\frac{-18±\sqrt{576}}{2\times 7}$$

Tuhia te pūtakerua o te 576.

$$x=\frac{-18±24}{2\times 7}$$

Whakareatia 2 ki te 7.

$$x=\frac{-18±24}{14}$$

Nā, me whakaoti te whārite x=\frac{-18±24}{14} ina he tāpiri te ±. Tāpiri -18 ki te 24.

$$x=\frac{6}{14}$$

Whakahekea te hautanga \frac{6}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{3}{7}$$

Nā, me whakaoti te whārite x=\frac{-18±24}{14} ina he tango te ±. Tango 24 mai i -18.

$$x=-\frac{42}{14}$$

Whakawehe -42 ki te 14.

$$x=-3$$

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{3}{7} mō te x_{1} me te -3 mō te x_{2}.

$$7x^{2}+18x-9=7\left(x-\frac{3}{7}\right)\left(x-\left(-3\right)\right)$$

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

$$7x^{2}+18x-9=7\left(x-\frac{3}{7}\right)\left(x+3\right)$$

Tango \frac{3}{7} 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.

$$7x^{2}+18x-9=7\times \frac{7x-3}{7}\left(x+3\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 7 i roto i te 7 me te 7.

$$7x^{2}+18x-9=\left(7x-3\right)\left(x+3\right)$$

Aromātai

$$\left(7x-3\right)\left(x+3\right)$$