Whakaoti i te 5r^2=11r+12

$$5 r ^ { 2 } = 11 r + 12$$

Whakaoti mō r

$$r=-\frac{4}{5}=-0.8$$

$$r=3$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Tangohia te 11r mai i ngā taha e rua.

$$5r^{2}-11r=12$$

Tangohia te 12 mai i ngā taha e rua.

$$5r^{2}-11r-12=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 5r^{2}+ar+br-12. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-11 ab=5\left(-12\right)=-60$$

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 -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 -11.

$$a=-15 b=4$$

Tuhia anō te 5r^{2}-11r-12 hei \left(5r^{2}-15r\right)+\left(4r-12\right).

$$\left(5r^{2}-15r\right)+\left(4r-12\right)$$

Tauwehea te 5r i te tuatahi me te 4 i te rōpū tuarua.

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

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

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

Hei kimi otinga whārite, me whakaoti te r-3=0 me te 5r+4=0.

$$r=3 r=-\frac{4}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 11r mai i ngā taha e rua.

$$5r^{2}-11r=12$$

Tangohia te 12 mai i ngā taha e rua.

$$5r^{2}-11r-12=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -11 mō b, me -12 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\times 5\left(-12\right)}}{2\times 5}$$

Pūrua -11.

$$r=\frac{-\left(-11\right)±\sqrt{121-4\times 5\left(-12\right)}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$r=\frac{-\left(-11\right)±\sqrt{121-20\left(-12\right)}}{2\times 5}$$

Whakareatia -20 ki te -12.

$$r=\frac{-\left(-11\right)±\sqrt{121+240}}{2\times 5}$$

Tāpiri 121 ki te 240.

$$r=\frac{-\left(-11\right)±\sqrt{361}}{2\times 5}$$

Tuhia te pūtakerua o te 361.

$$r=\frac{-\left(-11\right)±19}{2\times 5}$$

Ko te tauaro o -11 ko 11.

$$r=\frac{11±19}{2\times 5}$$

Whakareatia 2 ki te 5.

$$r=\frac{11±19}{10}$$

Nā, me whakaoti te whārite r=\frac{11±19}{10} ina he tāpiri te ±. Tāpiri 11 ki te 19.

$$r=\frac{30}{10}$$

Whakawehe 30 ki te 10.

$$r=3$$

Nā, me whakaoti te whārite r=\frac{11±19}{10} ina he tango te ±. Tango 19 mai i 11.

$$r=-\frac{8}{10}$$

Whakahekea te hautanga \frac{-8}{10} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$r=-\frac{4}{5}$$

Kua oti te whārite te whakatau.

$$r=3 r=-\frac{4}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 11r mai i ngā taha e rua.

$$5r^{2}-11r=12$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5r^{2}-11r}{5}=\frac{12}{5}$$

Mā te whakawehe ki te 5 ka wetekia te whakareanga ki te 5.

$$r^{2}-\frac{11}{5}r=\frac{12}{5}$$

Whakawehea te -\frac{11}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{10}. Nā, tāpiria te pūrua o te -\frac{11}{10} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$r^{2}-\frac{11}{5}r+\left(-\frac{11}{10}\right)^{2}=\frac{12}{5}+\left(-\frac{11}{10}\right)^{2}$$

Pūruatia -\frac{11}{10} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$r^{2}-\frac{11}{5}r+\frac{121}{100}=\frac{12}{5}+\frac{121}{100}$$

Tāpiri \frac{12}{5} ki te \frac{121}{100} 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.

$$r^{2}-\frac{11}{5}r+\frac{121}{100}=\frac{361}{100}$$

Tauwehea r^{2}-\frac{11}{5}r+\frac{121}{100}. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(r-\frac{11}{10}\right)^{2}=\frac{361}{100}$$

Tuhia te pūtakerua o ngā taha e rua o te whārite.

$$\sqrt{\left(r-\frac{11}{10}\right)^{2}}=\sqrt{\frac{361}{100}}$$

Whakarūnātia.

$$r-\frac{11}{10}=\frac{19}{10} r-\frac{11}{10}=-\frac{19}{10}$$

Me tāpiri \frac{11}{10} ki ngā taha e rua o te whārite.

$$r=3 r=-\frac{4}{5}$$