Whakaoti i te -8r^2+26r-15

$$- 8 r ^ { 2 } + 26 r - 15$$

Tauwehe

$$\left(3-4r\right)\left(2r-5\right)$$

Ngā Upane Otinga

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

$$a+b=26 ab=-8\left(-15\right)=120$$

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 120.

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

Tātaihia te tapeke mō ia takirua.

$$1+120=121 2+60=62 3+40=43 4+30=34 5+24=29 6+20=26 8+15=23 10+12=22$$

Ko te otinga te takirua ka hoatu i te tapeke 26.

$$a=20 b=6$$

Tuhia anō te -8r^{2}+26r-15 hei \left(-8r^{2}+20r\right)+\left(6r-15\right).

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

Tauwehea te -4r i te tuatahi me te 3 i te rōpū tuarua.

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

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

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

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.

$$-8r^{2}+26r-15=0$$

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.

$$r=\frac{-26±\sqrt{26^{2}-4\left(-8\right)\left(-15\right)}}{2\left(-8\right)}$$

Pūrua 26.

$$r=\frac{-26±\sqrt{676-4\left(-8\right)\left(-15\right)}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$r=\frac{-26±\sqrt{676+32\left(-15\right)}}{2\left(-8\right)}$$

Whakareatia 32 ki te -15.

$$r=\frac{-26±\sqrt{676-480}}{2\left(-8\right)}$$

Tāpiri 676 ki te -480.

$$r=\frac{-26±\sqrt{196}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 196.

$$r=\frac{-26±14}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$r=\frac{-26±14}{-16}$$

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

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

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

$$r=\frac{3}{4}$$

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

$$r=-\frac{40}{-16}$$

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

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

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

$$-8r^{2}+26r-15=-8\left(r-\frac{3}{4}\right)\left(r-\frac{5}{2}\right)$$

Tango \frac{3}{4} mai i r 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.

$$-8r^{2}+26r-15=-8\times \frac{-4r+3}{-4}\left(r-\frac{5}{2}\right)$$

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

$$-8r^{2}+26r-15=-8\times \frac{-4r+3}{-4}\times \frac{-2r+5}{-2}$$

Whakareatia \frac{-4r+3}{-4} ki te \frac{-2r+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.

$$-8r^{2}+26r-15=-8\times \frac{\left(-4r+3\right)\left(-2r+5\right)}{-4\left(-2\right)}$$

Whakareatia -4 ki te -2.

$$-8r^{2}+26r-15=-8\times \frac{\left(-4r+3\right)\left(-2r+5\right)}{8}$$

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

$$-8r^{2}+26r-15=-\left(-4r+3\right)\left(-2r+5\right)$$

Aromātai

$$\left(3-4r\right)\left(2r-5\right)$$