Whakaoti i te x^2/(4-x)(1-x)=13*10^9

$$\frac { x ^ { 2 } } { ( 4 - x ) ( 1 - x ) } = 13 \times 10 ^ { 9 }$$

Whakaoti mō x

$$x = \frac{20000 \sqrt{950625000130} + 32500000000}{12999999999} \approx 4$$

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}\approx 1$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Tātaihia te 10 mā te pū o 9, kia riro ko 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Whakareatia te 13 ki te 1000000000, ka 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 13000000000 ki te x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 13000000000x-52000000000 ki te x-1 ka whakakotahi i ngā kupu rite.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Tangohia te 13000000000x^{2} mai i ngā taha e rua.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Pahekotia te x^{2} me -13000000000x^{2}, ka -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Me tāpiri te 65000000000x ki ngā taha e rua.

$$-12999999999x^{2}[phone]x=52000000000$$

Tangohia te 52000000000 mai i ngā taha e rua.

$$-12999999999x^{2}[phone]x-52000000000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -12999999999 mō a, 65000000000 mō b, me -52000000000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-65000000000±\sqrt{65000000000^{2}-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Pūrua 65000000000.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-4\left(-12999999999\right)\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Whakareatia -4 ki te -12999999999.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000+51999999996\left(-52000000000\right)}}{2\left(-12999999999\right)}$$

Whakareatia 51999999996 ki te -52000000000.

$$x=\frac{-65000000000±\sqrt{4225000000000000000000-2703999999792000000000}}{2\left(-12999999999\right)}$$

Tāpiri 4225000000000000000000 ki te -2703999999792000000000.

$$x=\frac{-65000000000±\sqrt{1521000000208000000000}}{2\left(-12999999999\right)}$$

Tuhia te pūtakerua o te 1521000000208000000000.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{2\left(-12999999999\right)}$$

Whakareatia 2 ki te -12999999999.

$$x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998}$$

Nā, me whakaoti te whārite x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} ina he tāpiri te ±. Tāpiri -65000000000 ki te 40000\sqrt{950625000130}.

$$x=\frac{40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Whakawehe -65000000000+40000\sqrt{950625000130} ki te -25999999998.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$

Nā, me whakaoti te whārite x=\frac{-65000000000±40000\sqrt{950625000130}}{-25999999998} ina he tango te ±. Tango 40000\sqrt{950625000130} mai i -65000000000.

$$x=\frac{-40000\sqrt{950625000130}-65000000000}{-25999999998}$$

Whakawehe -65000000000-40000\sqrt{950625000130} ki te -25999999998.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Kua oti te whārite te whakatau.

$$x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999} x=\frac{20000\sqrt{950625000130}[phone]}{12999999999}$$

Ngā Upane mō te Whakaoti i te Pūrua

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

$$x^{2}=13\times 10^{9}\left(x-4\right)\left(x-1\right)$$

Tātaihia te 10 mā te pū o 9, kia riro ko 1000000000.

$$x^{2}=13\times 1000000000\left(x-4\right)\left(x-1\right)$$

Whakareatia te 13 ki te 1000000000, ka 13000000000.

$$x^{2}=13000000000\left(x-4\right)\left(x-1\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 13000000000 ki te x-4.

$$x^{2}=\left(13000000000x-52000000000\right)\left(x-1\right)$$

Whakamahia te āhuatanga tuaritanga hei whakarea te 13000000000x-52000000000 ki te x-1 ka whakakotahi i ngā kupu rite.

$$x^{2}=13000000000x^{2}-65000000000x+52000000000$$

Tangohia te 13000000000x^{2} mai i ngā taha e rua.

$$x^{2}-13000000000x^{2}=-65000000000x+52000000000$$

Pahekotia te x^{2} me -13000000000x^{2}, ka -12999999999x^{2}.

$$-12999999999x^{2}=-65000000000x+52000000000$$

Me tāpiri te 65000000000x ki ngā taha e rua.

$$-12999999999x^{2}[phone]x=52000000000$$

Whakawehea ngā taha e rua ki te -12999999999.

$$\frac{-12999999999x^{2}[phone]x}{-12999999999}=\frac{52000000000}{-12999999999}$$

Mā te whakawehe ki te -12999999999 ka wetekia te whakareanga ki te -12999999999.

$$x^{2}+\frac{65000000000}{-12999999999}x=\frac{52000000000}{-12999999999}$$

Whakawehe 65000000000 ki te -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=\frac{52000000000}{-12999999999}$$

Whakawehe 52000000000 ki te -12999999999.

$$x^{2}-\frac{65000000000}{12999999999}x=-\frac{52000000000}{12999999999}$$

Whakawehea te -\frac{65000000000}{12999999999}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{32500000000}{12999999999}. Nā, tāpiria te pūrua o te -\frac{32500000000}{12999999999} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{65000000000}{12999999999}x+\left(-\frac{32500000000}{12999999999}\right)^{2}=-\frac{52000000000}{12999999999}+\left(-\frac{32500000000}{12999999999}\right)^{2}$$

Pūruatia -\frac{32500000000}{12999999999} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=-\frac{52000000000}{12999999999}+\frac{1056250000000000000000}{168999999974000000001}$$

Tāpiri -\frac{52000000000}{12999999999} ki te \frac{1056250000000000000000}{168999999974000000001} 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.

$$x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}=\frac{380250000052000000000}{168999999974000000001}$$

Tauwehea x^{2}-\frac{65000000000}{12999999999}x+\frac{1056250000000000000000}{168999999974000000001}. 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}.

$$\left(x-\frac{32500000000}{12999999999}\right)^{2}=\frac{380250000052000000000}{168999999974000000001}$$

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

$$\sqrt{\left(x-\frac{32500000000}{12999999999}\right)^{2}}=\sqrt{\frac{380250000052000000000}{168999999974000000001}}$$

Whakarūnātia.

$$x-\frac{32500000000}{12999999999}=\frac{20000\sqrt{950625000130}}{12999999999} x-\frac{32500000000}{12999999999}=-\frac{20000\sqrt{950625000130}}{12999999999}$$

Me tāpiri \frac{32500000000}{12999999999} ki ngā taha e rua o te whārite.

$$x=\frac{20000\sqrt{950625000130}[phone]}{12999999999} x=\frac{32500000000-20000\sqrt{950625000130}}{12999999999}$$