Whakaoti i te (2x)^2/(1-x)(1-5-x)=12*10^-2

$$\frac { ( 2 x ) ^ { 2 } } { ( 1 - x ) ( 1 - 5 - x ) } = 12 \times 10 ^ { - 2 }$$

Whakaoti mō x (complex solution)

$$x=\frac{9+5\sqrt{183}i}{194}\approx 0.046391753+0.348653331i$$

$$x=\frac{-5\sqrt{183}i+9}{194}\approx 0.046391753-0.348653331i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -4,1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+4\right).

$$\left(2x\right)^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

Whakarohaina te \left(2x\right)^{2}.

$$2^{2}x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

Tātaihia te 10 mā te pū o -2, kia riro ko \frac{1}{100}.

$$4x^{2}=12\times \frac{1}{100}\left(x-1\right)\left(x+4\right)$$

Whakareatia te 12 ki te \frac{1}{100}, ka \frac{3}{25}.

$$4x^{2}=\frac{3}{25}\left(x-1\right)\left(x+4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{3}{25} ki te x-1.

$$4x^{2}=\left(\frac{3}{25}x-\frac{3}{25}\right)\left(x+4\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te \frac{3}{25}x-\frac{3}{25} ki te x+4 ka whakakotahi i ngā kupu rite.

$$4x^{2}=\frac{3}{25}x^{2}+\frac{9}{25}x-\frac{12}{25}$$

Tangohia te \frac{3}{25}x^{2} mai i ngā taha e rua.

$$4x^{2}-\frac{3}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

Pahekotia te 4x^{2} me -\frac{3}{25}x^{2}, ka \frac{97}{25}x^{2}.

$$\frac{97}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

Tangohia te \frac{9}{25}x mai i ngā taha e rua.

$$\frac{97}{25}x^{2}-\frac{9}{25}x=-\frac{12}{25}$$

Me tāpiri te \frac{12}{25} ki ngā taha e rua.

$$\frac{97}{25}x^{2}-\frac{9}{25}x+\frac{12}{25}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{97}{25} mō a, -\frac{9}{25} mō b, me \frac{12}{25} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\left(-\frac{9}{25}\right)^{2}-4\times \frac{97}{25}\times \frac{12}{25}}}{2\times \frac{97}{25}}$$

Pūruatia -\frac{9}{25} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\frac{81}{625}-4\times \frac{97}{25}\times \frac{12}{25}}}{2\times \frac{97}{25}}$$

Whakareatia -4 ki te \frac{97}{25}.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\frac{81}{625}-\frac{388}{25}\times \frac{12}{25}}}{2\times \frac{97}{25}}$$

Whakareatia -\frac{388}{25} ki te \frac{12}{25} 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.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{\frac{81-4656}{625}}}{2\times \frac{97}{25}}$$

Tāpiri \frac{81}{625} ki te -\frac{4656}{625} 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.

$$x=\frac{-\left(-\frac{9}{25}\right)±\sqrt{-\frac{183}{25}}}{2\times \frac{97}{25}}$$

Tuhia te pūtakerua o te -\frac{183}{25}.

$$x=\frac{-\left(-\frac{9}{25}\right)±\frac{\sqrt{183}i}{5}}{2\times \frac{97}{25}}$$

Ko te tauaro o -\frac{9}{25} ko \frac{9}{25}.

$$x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{2\times \frac{97}{25}}$$

Whakareatia 2 ki te \frac{97}{25}.

$$x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{\frac{194}{25}}$$

Nā, me whakaoti te whārite x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{\frac{194}{25}} ina he tāpiri te ±. Tāpiri \frac{9}{25} ki te \frac{i\sqrt{183}}{5}.

$$x=\frac{\frac{\sqrt{183}i}{5}+\frac{9}{25}}{\frac{194}{25}}$$

Whakawehe \frac{9}{25}+\frac{i\sqrt{183}}{5} ki te \frac{194}{25} mā te whakarea \frac{9}{25}+\frac{i\sqrt{183}}{5} ki te tau huripoki o \frac{194}{25}.

$$x=\frac{9+5\sqrt{183}i}{194}$$

Nā, me whakaoti te whārite x=\frac{\frac{9}{25}±\frac{\sqrt{183}i}{5}}{\frac{194}{25}} ina he tango te ±. Tango \frac{i\sqrt{183}}{5} mai i \frac{9}{25}.

$$x=\frac{-\frac{\sqrt{183}i}{5}+\frac{9}{25}}{\frac{194}{25}}$$

Whakawehe \frac{9}{25}-\frac{i\sqrt{183}}{5} ki te \frac{194}{25} mā te whakarea \frac{9}{25}-\frac{i\sqrt{183}}{5} ki te tau huripoki o \frac{194}{25}.

$$x=\frac{-5\sqrt{183}i+9}{194}$$

Kua oti te whārite te whakatau.

$$x=\frac{9+5\sqrt{183}i}{194} x=\frac{-5\sqrt{183}i+9}{194}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki tētahi o ngā uara -4,1 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te \left(x-1\right)\left(x+4\right).

$$\left(2x\right)^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

Whakarohaina te \left(2x\right)^{2}.

$$2^{2}x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

Tātaihia te 2 mā te pū o 2, kia riro ko 4.

$$4x^{2}=12\times 10^{-2}\left(x-1\right)\left(x+4\right)$$

Tātaihia te 10 mā te pū o -2, kia riro ko \frac{1}{100}.

$$4x^{2}=12\times \frac{1}{100}\left(x-1\right)\left(x+4\right)$$

Whakareatia te 12 ki te \frac{1}{100}, ka \frac{3}{25}.

$$4x^{2}=\frac{3}{25}\left(x-1\right)\left(x+4\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te \frac{3}{25} ki te x-1.

$$4x^{2}=\left(\frac{3}{25}x-\frac{3}{25}\right)\left(x+4\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te \frac{3}{25}x-\frac{3}{25} ki te x+4 ka whakakotahi i ngā kupu rite.

$$4x^{2}=\frac{3}{25}x^{2}+\frac{9}{25}x-\frac{12}{25}$$

Tangohia te \frac{3}{25}x^{2} mai i ngā taha e rua.

$$4x^{2}-\frac{3}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

Pahekotia te 4x^{2} me -\frac{3}{25}x^{2}, ka \frac{97}{25}x^{2}.

$$\frac{97}{25}x^{2}=\frac{9}{25}x-\frac{12}{25}$$

Tangohia te \frac{9}{25}x mai i ngā taha e rua.

$$\frac{97}{25}x^{2}-\frac{9}{25}x=-\frac{12}{25}$$

Whakawehea ngā taha e rua o te whārite ki te \frac{97}{25}, he ōrite ki te whakarea i ngā taha e rua ki te tau huripoki o te hautanga.

$$\frac{\frac{97}{25}x^{2}-\frac{9}{25}x}{\frac{97}{25}}=-\frac{\frac{12}{25}}{\frac{97}{25}}$$

Mā te whakawehe ki te \frac{97}{25} ka wetekia te whakareanga ki te \frac{97}{25}.

$$x^{2}+\left(-\frac{\frac{9}{25}}{\frac{97}{25}}\right)x=-\frac{\frac{12}{25}}{\frac{97}{25}}$$

Whakawehe -\frac{9}{25} ki te \frac{97}{25} mā te whakarea -\frac{9}{25} ki te tau huripoki o \frac{97}{25}.

$$x^{2}-\frac{9}{97}x=-\frac{\frac{12}{25}}{\frac{97}{25}}$$

Whakawehe -\frac{12}{25} ki te \frac{97}{25} mā te whakarea -\frac{12}{25} ki te tau huripoki o \frac{97}{25}.

$$x^{2}-\frac{9}{97}x=-\frac{12}{97}$$

Whakawehea te -\frac{9}{97}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{9}{194}. Nā, tāpiria te pūrua o te -\frac{9}{194} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{9}{97}x+\left(-\frac{9}{194}\right)^{2}=-\frac{12}{97}+\left(-\frac{9}{194}\right)^{2}$$

Pūruatia -\frac{9}{194} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{9}{97}x+\frac{81}{37636}=-\frac{12}{97}+\frac{81}{37636}$$

Tāpiri -\frac{12}{97} ki te \frac{81}{37636} 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.

$$x^{2}-\frac{9}{97}x+\frac{81}{37636}=-\frac{4575}{37636}$$

Tauwehea x^{2}-\frac{9}{97}x+\frac{81}{37636}. 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}.

$$\left(x-\frac{9}{194}\right)^{2}=-\frac{4575}{37636}$$

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

$$\sqrt{\left(x-\frac{9}{194}\right)^{2}}=\sqrt{-\frac{4575}{37636}}$$

Whakarūnātia.

$$x-\frac{9}{194}=\frac{5\sqrt{183}i}{194} x-\frac{9}{194}=-\frac{5\sqrt{183}i}{194}$$

Me tāpiri \frac{9}{194} ki ngā taha e rua o te whārite.

$$x=\frac{9+5\sqrt{183}i}{194} x=\frac{-5\sqrt{183}i+9}{194}$$