Whakaoti i te 9x^2-81x+50

$$9 { x }^{ 2 } -81x+50$$

Tauwehe

$$\left(3x-25\right)\left(3x-2\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 9x^{2}+ax+bx+50. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-81 ab=9\times 50=450$$

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 450.

$$-1,-450 -2,-225 -3,-150 -5,-90 -6,-75 -9,-50 -10,-45 -15,-30 -18,-25$$

Tātaihia te tapeke mō ia takirua.

$$-1-450=-451 -2-225=-227 -3-150=-153 -5-90=-95 -6-75=-81 -9-50=-59 -10-45=-55 -15-30=-45 -18-25=-43$$

Ko te otinga te takirua ka hoatu i te tapeke -81.

$$a=-75 b=-6$$

Tuhia anō te 9x^{2}-81x+50 hei \left(9x^{2}-75x\right)+\left(-6x+50\right).

$$\left(9x^{2}-75x\right)+\left(-6x+50\right)$$

Tauwehea te 3x i te tuatahi me te -2 i te rōpū tuarua.

$$3x\left(3x-25\right)-2\left(3x-25\right)$$

Whakatauwehea atu te kīanga pātahi 3x-25 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(3x-25\right)\left(3x-2\right)$$

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.

$$9x^{2}-81x+50=0$$

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.

$$x=\frac{-\left(-81\right)±\sqrt{\left(-81\right)^{2}-4\times 9\times 50}}{2\times 9}$$

Pūrua -81.

$$x=\frac{-\left(-81\right)±\sqrt{6561-4\times 9\times 50}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-81\right)±\sqrt{6561-36\times 50}}{2\times 9}$$

Whakareatia -36 ki te 50.

$$x=\frac{-\left(-81\right)±\sqrt{6561-1800}}{2\times 9}$$

Tāpiri 6561 ki te -1800.

$$x=\frac{-\left(-81\right)±\sqrt{4761}}{2\times 9}$$

Tuhia te pūtakerua o te 4761.

$$x=\frac{-\left(-81\right)±69}{2\times 9}$$

Ko te tauaro o -81 ko 81.

$$x=\frac{81±69}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{81±69}{18}$$

Nā, me whakaoti te whārite x=\frac{81±69}{18} ina he tāpiri te ±. Tāpiri 81 ki te 69.

$$x=\frac{150}{18}$$

Whakahekea te hautanga \frac{150}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{25}{3}$$

Nā, me whakaoti te whārite x=\frac{81±69}{18} ina he tango te ±. Tango 69 mai i 81.

$$x=\frac{12}{18}$$

Whakahekea te hautanga \frac{12}{18} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{2}{3}$$

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{25}{3} mō te x_{1} me te \frac{2}{3} mō te x_{2}.

$$9x^{2}-81x+50=9\left(x-\frac{25}{3}\right)\left(x-\frac{2}{3}\right)$$

Tango \frac{25}{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.

$$9x^{2}-81x+50=9\times \frac{3x-25}{3}\left(x-\frac{2}{3}\right)$$

Tango \frac{2}{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.

$$9x^{2}-81x+50=9\times \frac{3x-25}{3}\times \frac{3x-2}{3}$$

Whakareatia \frac{3x-25}{3} ki te \frac{3x-2}{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.

$$9x^{2}-81x+50=9\times \frac{\left(3x-25\right)\left(3x-2\right)}{3\times 3}$$

Whakareatia 3 ki te 3.

$$9x^{2}-81x+50=9\times \frac{\left(3x-25\right)\left(3x-2\right)}{9}$$

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

$$9x^{2}-81x+50=\left(3x-25\right)\left(3x-2\right)$$

Aromātai

$$\left(3x-25\right)\left(3x-2\right)$$