Whakaoti i te 2/b-3-6/2b+1=4

$$\frac{2}{b-3}-\frac{6}{2b+1}=4$$

Whakaoti mō b

$$b = \frac{\sqrt{337} + 9}{8} \approx 3.419694969$$

$$b=\frac{9-\sqrt{337}}{8}\approx -1.169694969$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe b ki tētahi o ngā uara -\frac{1}{2},3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(b-3\right)\left(2b+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o b-3,2b+1.

$$\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2b+1 ki te 2.

$$4b+2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te b-3 ki te 6.

$$4b+2-\left(6b-18\right)=4\left(b-3\right)\left(2b+1\right)$$

Hei kimi i te tauaro o 6b-18, kimihia te tauaro o ia taurangi.

$$4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)$$

Pahekotia te 4b me -6b, ka -2b.

$$-2b+2+18=4\left(b-3\right)\left(2b+1\right)$$

Tāpirihia te 2 ki te 18, ka 20.

$$-2b+20=4\left(b-3\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te b-3.

$$-2b+20=\left(4b-12\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4b-12 ki te 2b+1 ka whakakotahi i ngā kupu rite.

$$-2b+20=8b^{2}-20b-12$$

Tangohia te 8b^{2} mai i ngā taha e rua.

$$-2b+20-8b^{2}=-20b-12$$

Me tāpiri te 20b ki ngā taha e rua.

$$-2b+20-8b^{2}+20b=-12$$

Pahekotia te -2b me 20b, ka 18b.

$$18b+20-8b^{2}=-12$$

Me tāpiri te 12 ki ngā taha e rua.

$$18b+20-8b^{2}+12=0$$

Tāpirihia te 20 ki te 12, ka 32.

$$18b+32-8b^{2}=0$$

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.

$$-8b^{2}+18b+32=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -8 mō a, 18 mō b, me 32 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-18±\sqrt{18^{2}-4\left(-8\right)\times 32}}{2\left(-8\right)}$$

Pūrua 18.

$$b=\frac{-18±\sqrt{324-4\left(-8\right)\times 32}}{2\left(-8\right)}$$

Whakareatia -4 ki te -8.

$$b=\frac{-18±\sqrt{324+32\times 32}}{2\left(-8\right)}$$

Whakareatia 32 ki te 32.

$$b=\frac{-18±\sqrt{324+1024}}{2\left(-8\right)}$$

Tāpiri 324 ki te 1024.

$$b=\frac{-18±\sqrt{1348}}{2\left(-8\right)}$$

Tuhia te pūtakerua o te 1348.

$$b=\frac{-18±2\sqrt{337}}{2\left(-8\right)}$$

Whakareatia 2 ki te -8.

$$b=\frac{-18±2\sqrt{337}}{-16}$$

Nā, me whakaoti te whārite b=\frac{-18±2\sqrt{337}}{-16} ina he tāpiri te ±. Tāpiri -18 ki te 2\sqrt{337}.

$$b=\frac{2\sqrt{337}-18}{-16}$$

Whakawehe -18+2\sqrt{337} ki te -16.

$$b=\frac{9-\sqrt{337}}{8}$$

Nā, me whakaoti te whārite b=\frac{-18±2\sqrt{337}}{-16} ina he tango te ±. Tango 2\sqrt{337} mai i -18.

$$b=\frac{-2\sqrt{337}-18}{-16}$$

Whakawehe -18-2\sqrt{337} ki te -16.

$$b=\frac{\sqrt{337}+9}{8}$$

Kua oti te whārite te whakatau.

$$b=\frac{9-\sqrt{337}}{8} b=\frac{\sqrt{337}+9}{8}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe b ki tētahi o ngā uara -\frac{1}{2},3 nā te kore tautuhi i te whakawehenga mā te kore. Me whakarea ngā taha e rua o te whārite ki te \left(b-3\right)\left(2b+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o b-3,2b+1.

$$\left(2b+1\right)\times 2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2b+1 ki te 2.

$$4b+2-\left(b-3\right)\times 6=4\left(b-3\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te b-3 ki te 6.

$$4b+2-\left(6b-18\right)=4\left(b-3\right)\left(2b+1\right)$$

Hei kimi i te tauaro o 6b-18, kimihia te tauaro o ia taurangi.

$$4b+2-6b+18=4\left(b-3\right)\left(2b+1\right)$$

Pahekotia te 4b me -6b, ka -2b.

$$-2b+2+18=4\left(b-3\right)\left(2b+1\right)$$

Tāpirihia te 2 ki te 18, ka 20.

$$-2b+20=4\left(b-3\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te b-3.

$$-2b+20=\left(4b-12\right)\left(2b+1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 4b-12 ki te 2b+1 ka whakakotahi i ngā kupu rite.

$$-2b+20=8b^{2}-20b-12$$

Tangohia te 8b^{2} mai i ngā taha e rua.

$$-2b+20-8b^{2}=-20b-12$$

Me tāpiri te 20b ki ngā taha e rua.

$$-2b+20-8b^{2}+20b=-12$$

Pahekotia te -2b me 20b, ka 18b.

$$18b+20-8b^{2}=-12$$

Tangohia te 20 mai i ngā taha e rua.

$$18b-8b^{2}=-12-20$$

Tangohia te 20 i te -12, ka -32.

$$18b-8b^{2}=-32$$

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.

$$-8b^{2}+18b=-32$$

Whakawehea ngā taha e rua ki te -8.

$$\frac{-8b^{2}+18b}{-8}=-\frac{32}{-8}$$

Mā te whakawehe ki te -8 ka wetekia te whakareanga ki te -8.

$$b^{2}+\frac{18}{-8}b=-\frac{32}{-8}$$

Whakahekea te hautanga \frac{18}{-8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$b^{2}-\frac{9}{4}b=-\frac{32}{-8}$$

Whakawehe -32 ki te -8.

$$b^{2}-\frac{9}{4}b=4$$

Whakawehea te -\frac{9}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{8}. Nā, tāpiria te pūrua o te -\frac{9}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$b^{2}-\frac{9}{4}b+\left(-\frac{9}{8}\right)^{2}=4+\left(-\frac{9}{8}\right)^{2}$$

Pūruatia -\frac{9}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$b^{2}-\frac{9}{4}b+\frac{81}{64}=4+\frac{81}{64}$$

Tāpiri 4 ki te \frac{81}{64}.

$$b^{2}-\frac{9}{4}b+\frac{81}{64}=\frac{337}{64}$$

Tauwehea te b^{2}-\frac{9}{4}b+\frac{81}{64}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(b-\frac{9}{8}\right)^{2}=\frac{337}{64}$$

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

$$\sqrt{\left(b-\frac{9}{8}\right)^{2}}=\sqrt{\frac{337}{64}}$$

Whakarūnātia.

$$b-\frac{9}{8}=\frac{\sqrt{337}}{8} b-\frac{9}{8}=-\frac{\sqrt{337}}{8}$$

Me tāpiri \frac{9}{8} ki ngā taha e rua o te whārite.

$$b=\frac{\sqrt{337}+9}{8} b=\frac{9-\sqrt{337}}{8}$$