Whakaoti i te 16x^2+10x-9=

$$16 x ^ { 2 } + 10 x - 9 =$$

Tauwehe

$$\left(2x-1\right)\left(8x+9\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 16x^{2}+ax+bx-9. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=10 ab=16\left(-9\right)=-144$$

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 -144.

$$-1,144 -2,72 -3,48 -4,36 -6,24 -8,18 -9,16 -12,12$$

Tātaihia te tapeke mō ia takirua.

$$-1+144=143 -2+72=70 -3+48=45 -4+36=32 -6+24=18 -8+18=10 -9+16=7 -12+12=0$$

Ko te otinga te takirua ka hoatu i te tapeke 10.

$$a=-8 b=18$$

Tuhia anō te 16x^{2}+10x-9 hei \left(16x^{2}-8x\right)+\left(18x-9\right).

$$\left(16x^{2}-8x\right)+\left(18x-9\right)$$

Tauwehea te 8x i te tuatahi me te 9 i te rōpū tuarua.

$$8x\left(2x-1\right)+9\left(2x-1\right)$$

Whakatauwehea atu te kīanga pātahi 2x-1 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2x-1\right)\left(8x+9\right)$$

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.

$$16x^{2}+10x-9=0$$

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.

$$x=\frac{-10±\sqrt{10^{2}-4\times 16\left(-9\right)}}{2\times 16}$$

Pūrua 10.

$$x=\frac{-10±\sqrt{100-4\times 16\left(-9\right)}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$x=\frac{-10±\sqrt{100-64\left(-9\right)}}{2\times 16}$$

Whakareatia -64 ki te -9.

$$x=\frac{-10±\sqrt{100+576}}{2\times 16}$$

Tāpiri 100 ki te 576.

$$x=\frac{-10±\sqrt{676}}{2\times 16}$$

Tuhia te pūtakerua o te 676.

$$x=\frac{-10±26}{2\times 16}$$

Whakareatia 2 ki te 16.

$$x=\frac{-10±26}{32}$$

Nā, me whakaoti te whārite x=\frac{-10±26}{32} ina he tāpiri te ±. Tāpiri -10 ki te 26.

$$x=\frac{16}{32}$$

Whakahekea te hautanga \frac{16}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

$$x=\frac{1}{2}$$

Nā, me whakaoti te whārite x=\frac{-10±26}{32} ina he tango te ±. Tango 26 mai i -10.

$$x=-\frac{36}{32}$$

Whakahekea te hautanga \frac{-36}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$x=-\frac{9}{8}$$

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}{2} mō te x_{1} me te -\frac{9}{8} mō te x_{2}.

$$16x^{2}+10x-9=16\left(x-\frac{1}{2}\right)\left(x-\left(-\frac{9}{8}\right)\right)$$

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

$$16x^{2}+10x-9=16\left(x-\frac{1}{2}\right)\left(x+\frac{9}{8}\right)$$

Tango \frac{1}{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.

$$16x^{2}+10x-9=16\times \frac{2x-1}{2}\left(x+\frac{9}{8}\right)$$

Tāpiri \frac{9}{8} 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.

$$16x^{2}+10x-9=16\times \frac{2x-1}{2}\times \frac{8x+9}{8}$$

Whakareatia \frac{2x-1}{2} ki te \frac{8x+9}{8} 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.

$$16x^{2}+10x-9=16\times \frac{\left(2x-1\right)\left(8x+9\right)}{2\times 8}$$

Whakareatia 2 ki te 8.

$$16x^{2}+10x-9=16\times \frac{\left(2x-1\right)\left(8x+9\right)}{16}$$

Whakakorea atu te tauwehe pūnoa nui rawa 16 i roto i te 16 me te 16.

$$16x^{2}+10x-9=\left(2x-1\right)\left(8x+9\right)$$

Aromātai

$$\left(2x-1\right)\left(8x+9\right)$$