Whakaoti i te 23x^2+45x-2

$$23 x ^ { 2 } + 45 x - 2$$

Tauwehe

$$\left(23x-1\right)\left(x+2\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 23x^{2}+ax+bx-2. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=45 ab=23\left(-2\right)=-46$$

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 -46.

$$-1,46 -2,23$$

Tātaihia te tapeke mō ia takirua.

$$-1+46=45 -2+23=21$$

Ko te otinga te takirua ka hoatu i te tapeke 45.

$$a=-1 b=46$$

Tuhia anō te 23x^{2}+45x-2 hei \left(23x^{2}-x\right)+\left(46x-2\right).

$$\left(23x^{2}-x\right)+\left(46x-2\right)$$

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

$$x\left(23x-1\right)+2\left(23x-1\right)$$

Whakatauwehea atu te kīanga pātahi 23x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(23x-1\right)\left(x+2\right)$$

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.

$$23x^{2}+45x-2=0$$

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.

$$x=\frac{-45±\sqrt{45^{2}-4\times 23\left(-2\right)}}{2\times 23}$$

Pūrua 45.

$$x=\frac{-45±\sqrt{2025-4\times 23\left(-2\right)}}{2\times 23}$$

Whakareatia -4 ki te 23.

$$x=\frac{-45±\sqrt{2025-92\left(-2\right)}}{2\times 23}$$

Whakareatia -92 ki te -2.

$$x=\frac{-45±\sqrt{2025+184}}{2\times 23}$$

Tāpiri 2025 ki te 184.

$$x=\frac{-45±\sqrt{2209}}{2\times 23}$$

Tuhia te pūtakerua o te 2209.

$$x=\frac{-45±47}{2\times 23}$$

Whakareatia 2 ki te 23.

$$x=\frac{-45±47}{46}$$

Nā, me whakaoti te whārite x=\frac{-45±47}{46} ina he tāpiri te ±. Tāpiri -45 ki te 47.

$$x=\frac{2}{46}$$

Whakahekea te hautanga \frac{2}{46} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x=\frac{1}{23}$$

Nā, me whakaoti te whārite x=\frac{-45±47}{46} ina he tango te ±. Tango 47 mai i -45.

$$x=-\frac{92}{46}$$

Whakawehe -92 ki te 46.

$$x=-2$$

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{1}{23} mō te x_{1} me te -2 mō te x_{2}.

$$23x^{2}+45x-2=23\left(x-\frac{1}{23}\right)\left(x-\left(-2\right)\right)$$

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

$$23x^{2}+45x-2=23\left(x-\frac{1}{23}\right)\left(x+2\right)$$

Tango \frac{1}{23} 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.

$$23x^{2}+45x-2=23\times \frac{23x-1}{23}\left(x+2\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 23 i roto i te 23 me te 23.

$$23x^{2}+45x-2=\left(23x-1\right)\left(x+2\right)$$

Aromātai

$$\left(23x-1\right)\left(x+2\right)$$