Whakaoti i te 9x^2-6x-35

$$9 x ^ { 2 } - 6 x - 35$$

Tauwehe

$$\left(3x-7\right)\left(3x+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9x^{2}+ax+bx-35. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=9\left(-35\right)=-315$$

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 -315.

$$1,-315 3,-105 5,-63 7,-45 9,-35 15,-21$$

Tātaihia te tapeke mō ia takirua.

$$1-315=-314 3-105=-102 5-63=-58 7-45=-38 9-35=-26 15-21=-6$$

Ko te otinga te takirua ka hoatu i te tapeke -6.

$$a=-21 b=15$$

Tuhia anō te 9x^{2}-6x-35 hei \left(9x^{2}-21x\right)+\left(15x-35\right).

$$\left(9x^{2}-21x\right)+\left(15x-35\right)$$

Tauwehea te 3x i te tuatahi me te 5 i te rōpū tuarua.

$$3x\left(3x-7\right)+5\left(3x-7\right)$$

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

$$\left(3x-7\right)\left(3x+5\right)$$

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.

$$9x^{2}-6x-35=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 9\left(-35\right)}}{2\times 9}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 9\left(-35\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-6\right)±\sqrt{36-36\left(-35\right)}}{2\times 9}$$

Whakareatia -36 ki te -35.

$$x=\frac{-\left(-6\right)±\sqrt{36+1260}}{2\times 9}$$

Tāpiri 36 ki te 1260.

$$x=\frac{-\left(-6\right)±\sqrt{1296}}{2\times 9}$$

Tuhia te pūtakerua o te 1296.

$$x=\frac{-\left(-6\right)±36}{2\times 9}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±36}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{6±36}{18}$$

Nā, me whakaoti te whārite x=\frac{6±36}{18} ina he tāpiri te ±. Tāpiri 6 ki te 36.

$$x=\frac{42}{18}$$

Whakahekea te hautanga \frac{42}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

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

Nā, me whakaoti te whārite x=\frac{6±36}{18} ina he tango te ±. Tango 36 mai i 6.

$$x=-\frac{30}{18}$$

Whakahekea te hautanga \frac{-30}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=-\frac{5}{3}$$

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{7}{3} mō te x_{1} me te -\frac{5}{3} mō te x_{2}.

$$9x^{2}-6x-35=9\left(x-\frac{7}{3}\right)\left(x-\left(-\frac{5}{3}\right)\right)$$

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

$$9x^{2}-6x-35=9\left(x-\frac{7}{3}\right)\left(x+\frac{5}{3}\right)$$

Tango \frac{7}{3} 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.

$$9x^{2}-6x-35=9\times \frac{3x-7}{3}\left(x+\frac{5}{3}\right)$$

Tāpiri \frac{5}{3} 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.

$$9x^{2}-6x-35=9\times \frac{3x-7}{3}\times \frac{3x+5}{3}$$

Whakareatia \frac{3x-7}{3} ki te \frac{3x+5}{3} 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.

$$9x^{2}-6x-35=9\times \frac{\left(3x-7\right)\left(3x+5\right)}{3\times 3}$$

Whakareatia 3 ki te 3.

$$9x^{2}-6x-35=9\times \frac{\left(3x-7\right)\left(3x+5\right)}{9}$$

Whakakorea atu te tauwehe pūnoa nui rawa 9 i roto i te 9 me te 9.

$$9x^{2}-6x-35=\left(3x-7\right)\left(3x+5\right)$$

Aromātai

$$\left(3x-7\right)\left(3x+5\right)$$