Whakaoti i te -16y^2+148y-252

$$-16 { y }^{ 2 } +148y-252$$

Tauwehe

$$4\left(7-y\right)\left(4y-9\right)$$

Ngā Upane Otinga

Tauwehea te 4.

$$4\left(-4y^{2}+37y-63\right)$$

Whakaarohia te -4y^{2}+37y-63. Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei -4y^{2}+ay+by-63. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=37 ab=-4\left(-63\right)=252$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 252.

$$1,252 2,126 3,84 4,63 6,42 7,36 9,28 12,21 14,18$$

Tātaihia te tapeke mō ia takirua.

$$1+252=253 2+126=128 3+84=87 4+63=67 6+42=48 7+36=43 9+28=37 12+21=33 14+18=32$$

Ko te otinga te takirua ka hoatu i te tapeke 37.

$$a=28 b=9$$

Tuhia anō te -4y^{2}+37y-63 hei \left(-4y^{2}+28y\right)+\left(9y-63\right).

$$\left(-4y^{2}+28y\right)+\left(9y-63\right)$$

Tauwehea te 4y i te tuatahi me te -9 i te rōpū tuarua.

$$4y\left(-y+7\right)-9\left(-y+7\right)$$

Whakatauwehea atu te kīanga pātahi -y+7 mā te whakamahi i te āhuatanga tātai tohatoha.

$$\left(-y+7\right)\left(4y-9\right)$$

Me tuhi anō te kīanga whakatauwehe katoa.

$$4\left(-y+7\right)\left(4y-9\right)$$

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.

$$-16y^{2}+148y-252=0$$

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.

$$y=\frac{-148±\sqrt{148^{2}-4\left(-16\right)\left(-252\right)}}{2\left(-16\right)}$$

Pūrua 148.

$$y=\frac{-148±\sqrt{21904-4\left(-16\right)\left(-252\right)}}{2\left(-16\right)}$$

Whakareatia -4 ki te -16.

$$y=\frac{-148±\sqrt{21904+64\left(-252\right)}}{2\left(-16\right)}$$

Whakareatia 64 ki te -252.

$$y=\frac{-148±\sqrt{21904-16128}}{2\left(-16\right)}$$

Tāpiri 21904 ki te -16128.

$$y=\frac{-148±\sqrt{5776}}{2\left(-16\right)}$$

Tuhia te pūtakerua o te 5776.

$$y=\frac{-148±76}{2\left(-16\right)}$$

Whakareatia 2 ki te -16.

$$y=\frac{-148±76}{-32}$$

Nā, me whakaoti te whārite y=\frac{-148±76}{-32} ina he tāpiri te ±. Tāpiri -148 ki te 76.

$$y=-\frac{72}{-32}$$

Whakahekea te hautanga \frac{-72}{-32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 8.

$$y=\frac{9}{4}$$

Nā, me whakaoti te whārite y=\frac{-148±76}{-32} ina he tango te ±. Tango 76 mai i -148.

$$y=-\frac{224}{-32}$$

Whakawehe -224 ki te -32.

$$y=7$$

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{9}{4} mō te x_{1} me te 7 mō te x_{2}.

$$-16y^{2}+148y-252=-16\left(y-\frac{9}{4}\right)\left(y-7\right)$$

Tango \frac{9}{4} 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.

$$-16y^{2}+148y-252=-16\times \frac{-4y+9}{-4}\left(y-7\right)$$

Whakakorea atu te tauwehe pūnoa nui rawa 4 i roto i te -16 me te 4.

$$-16y^{2}+148y-252=4\left(-4y+9\right)\left(y-7\right)$$

Aromātai

$$-16y^{2}+148y-252$$