Whakaoti i te 4m^2+4m-15

$$4 m ^ { 2 } + 4 m - 15$$

Tauwehe

$$\left(2m-3\right)\left(2m+5\right)$$

Ngā Upane Otinga

Whakatauwehea te kīanga mā te whakarōpū. Tuatahi, me tuhi anō te kīanga hei 4m^{2}+am+bm-15. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=4 ab=4\left(-15\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Tātaihia te tapeke mō ia takirua.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

Ko te otinga te takirua ka hoatu i te tapeke 4.

$$a=-6 b=10$$

Tuhia anō te 4m^{2}+4m-15 hei \left(4m^{2}-6m\right)+\left(10m-15\right).

$$\left(4m^{2}-6m\right)+\left(10m-15\right)$$

Tauwehea te 2m i te tuatahi me te 5 i te rōpū tuarua.

$$2m\left(2m-3\right)+5\left(2m-3\right)$$

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

$$\left(2m-3\right)\left(2m+5\right)$$

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.

$$4m^{2}+4m-15=0$$

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.

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

Pūrua 4.

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

Whakareatia -4 ki te 4.

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

Whakareatia -16 ki te -15.

$$m=\frac{-4±\sqrt{16+240}}{2\times 4}$$

Tāpiri 16 ki te 240.

$$m=\frac{-4±\sqrt{256}}{2\times 4}$$

Tuhia te pūtakerua o te 256.

$$m=\frac{-4±16}{2\times 4}$$

Whakareatia 2 ki te 4.

$$m=\frac{-4±16}{8}$$

Nā, me whakaoti te whārite m=\frac{-4±16}{8} ina he tāpiri te ±. Tāpiri -4 ki te 16.

$$m=\frac{12}{8}$$

Whakahekea te hautanga \frac{12}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

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

Nā, me whakaoti te whārite m=\frac{-4±16}{8} ina he tango te ±. Tango 16 mai i -4.

$$m=-\frac{20}{8}$$

Whakahekea te hautanga \frac{-20}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$m=-\frac{5}{2}$$

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{5}{2} mō te x_{2}.

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

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

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

Tango \frac{3}{2} mai i m 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.

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

Tāpiri \frac{5}{2} ki te m 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.

$$4m^{2}+4m-15=4\times \frac{2m-3}{2}\times \frac{2m+5}{2}$$

Whakareatia \frac{2m-3}{2} ki te \frac{2m+5}{2} 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.

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

Whakareatia 2 ki te 2.

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

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

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

Aromātai

$$\left(2m-3\right)\left(2m+5\right)$$