Whakaoti i te 30b^2-28b-16=0

$$30 b ^ { 2 } - 28 b - 16 = 0$$

Whakaoti mō b

$$b=-\frac{2}{5}=-0.4$$

$$b = \frac{4}{3} = 1\frac{1}{3} \approx 1.333333333$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

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

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 15b^{2}+ab+bb-8. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

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

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 15b^{2}-14b-8 hei \left(15b^{2}-20b\right)+\left(6b-8\right).

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

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

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

Whakatauwehea atu te kīanga pātahi 3b-4 mā te whakamahi i te āhuatanga tātai tohatoha.

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

Hei kimi otinga whārite, me whakaoti te 3b-4=0 me te 5b+2=0.

$$b=\frac{4}{3} b=-\frac{2}{5}$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$30b^{2}-28b-16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 30 mō a, -28 mō b, me -16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 30\left(-16\right)}}{2\times 30}$$

Pūrua -28.

$$b=\frac{-\left(-28\right)±\sqrt{784-4\times 30\left(-16\right)}}{2\times 30}$$

Whakareatia -4 ki te 30.

$$b=\frac{-\left(-28\right)±\sqrt{784-120\left(-16\right)}}{2\times 30}$$

Whakareatia -120 ki te -16.

$$b=\frac{-\left(-28\right)±\sqrt{784+1920}}{2\times 30}$$

Tāpiri 784 ki te 1920.

$$b=\frac{-\left(-28\right)±\sqrt{2704}}{2\times 30}$$

Tuhia te pūtakerua o te 2704.

$$b=\frac{-\left(-28\right)±52}{2\times 30}$$

Ko te tauaro o -28 ko 28.

$$b=\frac{28±52}{2\times 30}$$

Whakareatia 2 ki te 30.

$$b=\frac{28±52}{60}$$

Nā, me whakaoti te whārite b=\frac{28±52}{60} ina he tāpiri te ±. Tāpiri 28 ki te 52.

$$b=\frac{80}{60}$$

Whakahekea te hautanga \frac{80}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 20.

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

Nā, me whakaoti te whārite b=\frac{28±52}{60} ina he tango te ±. Tango 52 mai i 28.

$$b=-\frac{24}{60}$$

Whakahekea te hautanga \frac{-24}{60} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 12.

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

Kua oti te whārite te whakatau.

$$b=\frac{4}{3} b=-\frac{2}{5}$$

Ngā Upane mō te Whakaoti i te Pūrua

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$30b^{2}-28b-16=0$$

Me tāpiri 16 ki ngā taha e rua o te whārite.

$$30b^{2}-28b-16-\left(-16\right)=-\left(-16\right)$$

Mā te tango i te -16 i a ia ake anō ka toe ko te 0.

$$30b^{2}-28b=-\left(-16\right)$$

Tango -16 mai i 0.

$$30b^{2}-28b=16$$

Whakawehea ngā taha e rua ki te 30.

$$\frac{30b^{2}-28b}{30}=\frac{16}{30}$$

Mā te whakawehe ki te 30 ka wetekia te whakareanga ki te 30.

$$b^{2}+\left(-\frac{28}{30}\right)b=\frac{16}{30}$$

Whakahekea te hautanga \frac{-28}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$b^{2}-\frac{14}{15}b=\frac{16}{30}$$

Whakahekea te hautanga \frac{16}{30} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$b^{2}-\frac{14}{15}b=\frac{8}{15}$$

Whakawehea te -\frac{14}{15}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{15}. Nā, tāpiria te pūrua o te -\frac{7}{15} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$b^{2}-\frac{14}{15}b+\left(-\frac{7}{15}\right)^{2}=\frac{8}{15}+\left(-\frac{7}{15}\right)^{2}$$

Pūruatia -\frac{7}{15} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$b^{2}-\frac{14}{15}b+\frac{49}{225}=\frac{8}{15}+\frac{49}{225}$$

Tāpiri \frac{8}{15} ki te \frac{49}{225} 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.

$$b^{2}-\frac{14}{15}b+\frac{49}{225}=\frac{169}{225}$$

Tauwehea b^{2}-\frac{14}{15}b+\frac{49}{225}. 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}.

$$\left(b-\frac{7}{15}\right)^{2}=\frac{169}{225}$$

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

$$\sqrt{\left(b-\frac{7}{15}\right)^{2}}=\sqrt{\frac{169}{225}}$$

Whakarūnātia.

$$b-\frac{7}{15}=\frac{13}{15} b-\frac{7}{15}=-\frac{13}{15}$$

Me tāpiri \frac{7}{15} ki ngā taha e rua o te whārite.

$$b=\frac{4}{3} b=-\frac{2}{5}$$