Whakaoti i te 8y^2-14y-15=

$$8 y ^ { 2 } - 14 y - 15 =$$

Tauwehe

$$\left(2y-5\right)\left(4y+3\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 8y^{2}+ay+by-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-14 ab=8\left(-15\right)=-120$$

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 -120.

$$1,-120 2,-60 3,-40 4,-30 5,-24 6,-20 8,-15 10,-12$$

Tātaihia te tapeke mō ia takirua.

$$1-120=-119 2-60=-58 3-40=-37 4-30=-26 5-24=-19 6-20=-14 8-15=-7 10-12=-2$$

Ko te otinga te takirua ka hoatu i te tapeke -14.

$$a=-20 b=6$$

Tuhia anō te 8y^{2}-14y-15 hei \left(8y^{2}-20y\right)+\left(6y-15\right).

$$\left(8y^{2}-20y\right)+\left(6y-15\right)$$

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

$$4y\left(2y-5\right)+3\left(2y-5\right)$$

Whakatauwehea atu te kīanga pātahi 2y-5 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(2y-5\right)\left(4y+3\right)$$

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.

$$8y^{2}-14y-15=0$$

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.

$$y=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 8\left(-15\right)}}{2\times 8}$$

Pūrua -14.

$$y=\frac{-\left(-14\right)±\sqrt{196-4\times 8\left(-15\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$y=\frac{-\left(-14\right)±\sqrt{196-32\left(-15\right)}}{2\times 8}$$

Whakareatia -32 ki te -15.

$$y=\frac{-\left(-14\right)±\sqrt{196+480}}{2\times 8}$$

Tāpiri 196 ki te 480.

$$y=\frac{-\left(-14\right)±\sqrt{676}}{2\times 8}$$

Tuhia te pūtakerua o te 676.

$$y=\frac{-\left(-14\right)±26}{2\times 8}$$

Ko te tauaro o -14 ko 14.

$$y=\frac{14±26}{2\times 8}$$

Whakareatia 2 ki te 8.

$$y=\frac{14±26}{16}$$

Nā, me whakaoti te whārite y=\frac{14±26}{16} ina he tāpiri te ±. Tāpiri 14 ki te 26.

$$y=\frac{40}{16}$$

Whakahekea te hautanga \frac{40}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$y=\frac{5}{2}$$

Nā, me whakaoti te whārite y=\frac{14±26}{16} ina he tango te ±. Tango 26 mai i 14.

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

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

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

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{5}{2} mō te x_{1} me te -\frac{3}{4} mō te x_{2}.

$$8y^{2}-14y-15=8\left(y-\frac{5}{2}\right)\left(y-\left(-\frac{3}{4}\right)\right)$$

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

$$8y^{2}-14y-15=8\left(y-\frac{5}{2}\right)\left(y+\frac{3}{4}\right)$$

Tango \frac{5}{2} mai i y 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.

$$8y^{2}-14y-15=8\times \frac{2y-5}{2}\left(y+\frac{3}{4}\right)$$

Tāpiri \frac{3}{4} ki te y 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.

$$8y^{2}-14y-15=8\times \frac{2y-5}{2}\times \frac{4y+3}{4}$$

Whakareatia \frac{2y-5}{2} ki te \frac{4y+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.

$$8y^{2}-14y-15=8\times \frac{\left(2y-5\right)\left(4y+3\right)}{2\times 4}$$

Whakareatia 2 ki te 4.

$$8y^{2}-14y-15=8\times \frac{\left(2y-5\right)\left(4y+3\right)}{8}$$

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

$$8y^{2}-14y-15=\left(2y-5\right)\left(4y+3\right)$$

Aromātai

$$\left(2y-5\right)\left(4y+3\right)$$