Whakaoti i te 8x^2-6x-9

$$8 x ^ { 2 } - 6 x - 9$$

Tauwehe

$$\left(2x-3\right)\left(4x+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 8x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-6 ab=8\left(-9\right)=-72$$

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 -72.

$$1,-72 2,-36 3,-24 4,-18 6,-12 8,-9$$

Tātaihia te tapeke mō ia takirua.

$$1-72=-71 2-36=-34 3-24=-21 4-18=-14 6-12=-6 8-9=-1$$

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

$$a=-12 b=6$$

Tuhia anō te 8x^{2}-6x-9 hei \left(8x^{2}-12x\right)+\left(6x-9\right).

$$\left(8x^{2}-12x\right)+\left(6x-9\right)$$

Tauwehea te 4x i te tuatahi me te 3 i te rōpū tuarua.

$$4x\left(2x-3\right)+3\left(2x-3\right)$$

Whakatauwehea atu te kīanga pātahi 2x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-3\right)\left(4x+3\right)$$

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.

$$8x^{2}-6x-9=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 8\left(-9\right)}}{2\times 8}$$

Pūrua -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 8\left(-9\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-6\right)±\sqrt{36-32\left(-9\right)}}{2\times 8}$$

Whakareatia -32 ki te -9.

$$x=\frac{-\left(-6\right)±\sqrt{36+288}}{2\times 8}$$

Tāpiri 36 ki te 288.

$$x=\frac{-\left(-6\right)±\sqrt{324}}{2\times 8}$$

Tuhia te pūtakerua o te 324.

$$x=\frac{-\left(-6\right)±18}{2\times 8}$$

Ko te tauaro o -6 ko 6.

$$x=\frac{6±18}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{6±18}{16}$$

Nā, me whakaoti te whārite x=\frac{6±18}{16} ina he tāpiri te ±. Tāpiri 6 ki te 18.

$$x=\frac{24}{16}$$

Whakahekea te hautanga \frac{24}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

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

Nā, me whakaoti te whārite x=\frac{6±18}{16} ina he tango te ±. Tango 18 mai i 6.

$$x=-\frac{12}{16}$$

Whakahekea te hautanga \frac{-12}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{3}{4}$$

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}{2} mō te x_{1} me te -\frac{3}{4} mō te x_{2}.

$$8x^{2}-6x-9=8\left(x-\frac{3}{2}\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$8x^{2}-6x-9=8\left(x-\frac{3}{2}\right)\left(x+\frac{3}{4}\right)$$

Tango \frac{3}{2} 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.

$$8x^{2}-6x-9=8\times \frac{2x-3}{2}\left(x+\frac{3}{4}\right)$$

Tāpiri \frac{3}{4} 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.

$$8x^{2}-6x-9=8\times \frac{2x-3}{2}\times \frac{4x+3}{4}$$

Whakareatia \frac{2x-3}{2} ki te \frac{4x+3}{4} 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.

$$8x^{2}-6x-9=8\times \frac{\left(2x-3\right)\left(4x+3\right)}{2\times 4}$$

Whakareatia 2 ki te 4.

$$8x^{2}-6x-9=8\times \frac{\left(2x-3\right)\left(4x+3\right)}{8}$$

Whakakorea atu te tauwehe pūnoa nui rawa 8 i roto i te 8 me te 8.

$$8x^{2}-6x-9=\left(2x-3\right)\left(4x+3\right)$$

Aromātai

$$\left(2x-3\right)\left(4x+3\right)$$