Whakaoti i te 10x^2-89x-9

$$10 x ^ { 2 } - 89 x - 9$$

Tauwehe

$$\left(x-9\right)\left(10x+1\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 10x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-89 ab=10\left(-9\right)=-90$$

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 -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Tātaihia te tapeke mō ia takirua.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

Ko te otinga te takirua ka hoatu i te tapeke -89.

$$a=-90 b=1$$

Tuhia anō te 10x^{2}-89x-9 hei \left(10x^{2}-90x\right)+\left(x-9\right).

$$\left(10x^{2}-90x\right)+\left(x-9\right)$$

Whakatauwehea atu 10x i te 10x^{2}-90x.

$$10x\left(x-9\right)+x-9$$

Whakatauwehea atu te kīanga pātahi x-9 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(x-9\right)\left(10x+1\right)$$

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.

$$10x^{2}-89x-9=0$$

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.

$$x=\frac{-\left(-89\right)±\sqrt{\left(-89\right)^{2}-4\times 10\left(-9\right)}}{2\times 10}$$

Pūrua -89.

$$x=\frac{-\left(-89\right)±\sqrt{7921-4\times 10\left(-9\right)}}{2\times 10}$$

Whakareatia -4 ki te 10.

$$x=\frac{-\left(-89\right)±\sqrt{7921-40\left(-9\right)}}{2\times 10}$$

Whakareatia -40 ki te -9.

$$x=\frac{-\left(-89\right)±\sqrt{7921+360}}{2\times 10}$$

Tāpiri 7921 ki te 360.

$$x=\frac{-\left(-89\right)±\sqrt{8281}}{2\times 10}$$

Tuhia te pūtakerua o te 8281.

$$x=\frac{-\left(-89\right)±91}{2\times 10}$$

Ko te tauaro o -89 ko 89.

$$x=\frac{89±91}{2\times 10}$$

Whakareatia 2 ki te 10.

$$x=\frac{89±91}{20}$$

Nā, me whakaoti te whārite x=\frac{89±91}{20} ina he tāpiri te ±. Tāpiri 89 ki te 91.

$$x=\frac{180}{20}$$

Whakawehe 180 ki te 20.

$$x=9$$

Nā, me whakaoti te whārite x=\frac{89±91}{20} ina he tango te ±. Tango 91 mai i 89.

$$x=-\frac{2}{20}$$

Whakahekea te hautanga \frac{-2}{20} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=-\frac{1}{10}$$

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 9 mō te x_{1} me te -\frac{1}{10} mō te x_{2}.

$$10x^{2}-89x-9=10\left(x-9\right)\left(x-\left(-\frac{1}{10}\right)\right)$$

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

$$10x^{2}-89x-9=10\left(x-9\right)\left(x+\frac{1}{10}\right)$$

Tāpiri \frac{1}{10} 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.

$$10x^{2}-89x-9=10\left(x-9\right)\times \frac{10x+1}{10}$$

Whakakorea atu te tauwehe pūnoa nui rawa 10 i roto i te 10 me te 10.

$$10x^{2}-89x-9=\left(x-9\right)\left(10x+1\right)$$

Aromātai

$$\left(x-9\right)\left(10x+1\right)$$