Whakaoti i te 15x^2+16x-15=

$$15 x ^ { 2 } + 16 x - 15 =$$

Tauwehe

$$\left(5x-3\right)\left(3x+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 15x^{2}+ax+bx-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=16 ab=15\left(-15\right)=-225$$

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 -225.

$$-1,225 -3,75 -5,45 -9,25 -15,15$$

Tātaihia te tapeke mō ia takirua.

$$-1+225=224 -3+75=72 -5+45=40 -9+25=16 -15+15=0$$

Ko te otinga te takirua ka hoatu i te tapeke 16.

$$a=-9 b=25$$

Tuhia anō te 15x^{2}+16x-15 hei \left(15x^{2}-9x\right)+\left(25x-15\right).

$$\left(15x^{2}-9x\right)+\left(25x-15\right)$$

Tauwehea te 3x i te tuatahi me te 5 i te rōpū tuarua.

$$3x\left(5x-3\right)+5\left(5x-3\right)$$

Whakatauwehea atu te kīanga pātahi 5x-3 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(5x-3\right)\left(3x+5\right)$$

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.

$$15x^{2}+16x-15=0$$

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.

$$x=\frac{-16±\sqrt{16^{2}-4\times 15\left(-15\right)}}{2\times 15}$$

Pūrua 16.

$$x=\frac{-16±\sqrt{256-4\times 15\left(-15\right)}}{2\times 15}$$

Whakareatia -4 ki te 15.

$$x=\frac{-16±\sqrt{256-60\left(-15\right)}}{2\times 15}$$

Whakareatia -60 ki te -15.

$$x=\frac{-16±\sqrt{256+900}}{2\times 15}$$

Tāpiri 256 ki te 900.

$$x=\frac{-16±\sqrt{1156}}{2\times 15}$$

Tuhia te pūtakerua o te 1156.

$$x=\frac{-16±34}{2\times 15}$$

Whakareatia 2 ki te 15.

$$x=\frac{-16±34}{30}$$

Nā, me whakaoti te whārite x=\frac{-16±34}{30} ina he tāpiri te ±. Tāpiri -16 ki te 34.

$$x=\frac{18}{30}$$

Whakahekea te hautanga \frac{18}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 6.

$$x=\frac{3}{5}$$

Nā, me whakaoti te whārite x=\frac{-16±34}{30} ina he tango te ±. Tango 34 mai i -16.

$$x=-\frac{50}{30}$$

Whakahekea te hautanga \frac{-50}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 10.

$$x=-\frac{5}{3}$$

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{3}{5} mō te x_{1} me te -\frac{5}{3} mō te x_{2}.

$$15x^{2}+16x-15=15\left(x-\frac{3}{5}\right)\left(x-\left(-\frac{5}{3}\right)\right)$$

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

$$15x^{2}+16x-15=15\left(x-\frac{3}{5}\right)\left(x+\frac{5}{3}\right)$$

Tango \frac{3}{5} 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.

$$15x^{2}+16x-15=15\times \frac{5x-3}{5}\left(x+\frac{5}{3}\right)$$

Tāpiri \frac{5}{3} 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.

$$15x^{2}+16x-15=15\times \frac{5x-3}{5}\times \frac{3x+5}{3}$$

Whakareatia \frac{5x-3}{5} ki te \frac{3x+5}{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.

$$15x^{2}+16x-15=15\times \frac{\left(5x-3\right)\left(3x+5\right)}{5\times 3}$$

Whakareatia 5 ki te 3.

$$15x^{2}+16x-15=15\times \frac{\left(5x-3\right)\left(3x+5\right)}{15}$$

Whakakorea atu te tauwehe pūnoa nui rawa 15 i roto i te 15 me te 15.

$$15x^{2}+16x-15=\left(5x-3\right)\left(3x+5\right)$$

Aromātai

$$\left(5x-3\right)\left(3x+5\right)$$