Whakaoti i te 16b^2-44b+10=0

$$16 b ^ { 2 } - 44 b + 10 = 0$$

Whakaoti mō b

$$b=\frac{1}{4}=0.25$$

$$b = \frac{5}{2} = 2\frac{1}{2} = 2.5$$

Ngā Upane Whakamahi Tauwehe Mā te Whakarōpū

Whakawehea ngā taha e rua ki te 2.

$$8b^{2}-22b+5=0$$

Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 8b^{2}+ab+bb+5. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.

$$a+b=-22 ab=8\times 5=40$$

I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōraro te a+b, he tōraro hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 40.

$$-1,-40 -2,-20 -4,-10 -5,-8$$

Tātaihia te tapeke mō ia takirua.

$$-1-40=-41 -2-20=-22 -4-10=-14 -5-8=-13$$

Ko te otinga te takirua ka hoatu i te tapeke -22.

$$a=-20 b=-2$$

Tuhia anō te 8b^{2}-22b+5 hei \left(8b^{2}-20b\right)+\left(-2b+5\right).

$$\left(8b^{2}-20b\right)+\left(-2b+5\right)$$

Tauwehea te 4b i te tuatahi me te -1 i te rōpū tuarua.

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

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

$$\left(2b-5\right)\left(4b-1\right)$$

Hei kimi otinga whārite, me whakaoti te 2b-5=0 me te 4b-1=0.

$$b=\frac{5}{2} b=\frac{1}{4}$$

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.

$$16b^{2}-44b+10=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 16 mō a, -44 mō b, me 10 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-\left(-44\right)±\sqrt{\left(-44\right)^{2}-4\times 16\times 10}}{2\times 16}$$

Pūrua -44.

$$b=\frac{-\left(-44\right)±\sqrt{1936-4\times 16\times 10}}{2\times 16}$$

Whakareatia -4 ki te 16.

$$b=\frac{-\left(-44\right)±\sqrt{1936-64\times 10}}{2\times 16}$$

Whakareatia -64 ki te 10.

$$b=\frac{-\left(-44\right)±\sqrt{1936-640}}{2\times 16}$$

Tāpiri 1936 ki te -640.

$$b=\frac{-\left(-44\right)±\sqrt{1296}}{2\times 16}$$

Tuhia te pūtakerua o te 1296.

$$b=\frac{-\left(-44\right)±36}{2\times 16}$$

Ko te tauaro o -44 ko 44.

$$b=\frac{44±36}{2\times 16}$$

Whakareatia 2 ki te 16.

$$b=\frac{44±36}{32}$$

Nā, me whakaoti te whārite b=\frac{44±36}{32} ina he tāpiri te ±. Tāpiri 44 ki te 36.

$$b=\frac{80}{32}$$

Whakahekea te hautanga \frac{80}{32} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 16.

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

Nā, me whakaoti te whārite b=\frac{44±36}{32} ina he tango te ±. Tango 36 mai i 44.

$$b=\frac{8}{32}$$

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

$$b=\frac{1}{4}$$

Kua oti te whārite te whakatau.

$$b=\frac{5}{2} b=\frac{1}{4}$$

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.

$$16b^{2}-44b+10=0$$

Me tango 10 mai i ngā taha e rua o te whārite.

$$16b^{2}-44b+10-10=-10$$

Mā te tango i te 10 i a ia ake anō ka toe ko te 0.

$$16b^{2}-44b=-10$$

Whakawehea ngā taha e rua ki te 16.

$$\frac{16b^{2}-44b}{16}=-\frac{10}{16}$$

Mā te whakawehe ki te 16 ka wetekia te whakareanga ki te 16.

$$b^{2}+\left(-\frac{44}{16}\right)b=-\frac{10}{16}$$

Whakahekea te hautanga \frac{-44}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.

$$b^{2}-\frac{11}{4}b=-\frac{10}{16}$$

Whakahekea te hautanga \frac{-10}{16} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$b^{2}-\frac{11}{4}b=-\frac{5}{8}$$

Whakawehea te -\frac{11}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{11}{8}. Nā, tāpiria te pūrua o te -\frac{11}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$b^{2}-\frac{11}{4}b+\left(-\frac{11}{8}\right)^{2}=-\frac{5}{8}+\left(-\frac{11}{8}\right)^{2}$$

Pūruatia -\frac{11}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$b^{2}-\frac{11}{4}b+\frac{121}{64}=-\frac{5}{8}+\frac{121}{64}$$

Tāpiri -\frac{5}{8} ki te \frac{121}{64} 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.

$$b^{2}-\frac{11}{4}b+\frac{121}{64}=\frac{81}{64}$$

Tauwehea b^{2}-\frac{11}{4}b+\frac{121}{64}. 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}.

$$\left(b-\frac{11}{8}\right)^{2}=\frac{81}{64}$$

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

$$\sqrt{\left(b-\frac{11}{8}\right)^{2}}=\sqrt{\frac{81}{64}}$$

Whakarūnātia.

$$b-\frac{11}{8}=\frac{9}{8} b-\frac{11}{8}=-\frac{9}{8}$$

Me tāpiri \frac{11}{8} ki ngā taha e rua o te whārite.

$$b=\frac{5}{2} b=\frac{1}{4}$$