Whakaoti i te (100+x)(100+x)1/x=204

$$(100+x)(100+x) \frac{ 1 }{ x } =204$$

Whakaoti mō x (complex solution)

$$x=2+14\sqrt{51}i\approx 2+99.979998i$$

$$x=-14\sqrt{51}i+2\approx 2-99.979998i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$\left(100+x\right)\left(100+x\right)\times 1=204x$$

Whakareatia te 100+x ki te 100+x, ka \left(100+x\right)^{2}.

$$\left(100+x\right)^{2}\times 1=204x$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(100+x\right)^{2}.

$$\left(10000+200x+x^{2}\right)\times 1=204x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10000+200x+x^{2} ki te 1.

$$10000+200x+x^{2}=204x$$

Tangohia te 204x mai i ngā taha e rua.

$$10000+200x+x^{2}-204x=0$$

Pahekotia te 200x me -204x, ka -4x.

$$10000-4x+x^{2}=0$$

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.

$$x^{2}-4x+10000=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -4 mō b, me 10000 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 10000}}{2}$$

Pūrua -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 10000}}{2}$$

Whakareatia -4 ki te 10000.

$$x=\frac{-\left(-4\right)±\sqrt{16-40000}}{2}$$

Tāpiri 16 ki te -40000.

$$x=\frac{-\left(-4\right)±\sqrt{-39984}}{2}$$

Tuhia te pūtakerua o te -39984.

$$x=\frac{-\left(-4\right)±28\sqrt{51}i}{2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±28\sqrt{51}i}{2}$$

Nā, me whakaoti te whārite x=\frac{4±28\sqrt{51}i}{2} ina he tāpiri te ±. Tāpiri 4 ki te 28i\sqrt{51}.

$$x=\frac{4+28\sqrt{51}i}{2}$$

Whakawehe 4+28i\sqrt{51} ki te 2.

$$x=2+14\sqrt{51}i$$

Nā, me whakaoti te whārite x=\frac{4±28\sqrt{51}i}{2} ina he tango te ±. Tango 28i\sqrt{51} mai i 4.

$$x=\frac{-28\sqrt{51}i+4}{2}$$

Whakawehe 4-28i\sqrt{51} ki te 2.

$$x=-14\sqrt{51}i+2$$

Kua oti te whārite te whakatau.

$$x=2+14\sqrt{51}i x=-14\sqrt{51}i+2$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$\left(100+x\right)\left(100+x\right)\times 1=204x$$

Whakareatia te 100+x ki te 100+x, ka \left(100+x\right)^{2}.

$$\left(100+x\right)^{2}\times 1=204x$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(100+x\right)^{2}.

$$\left(10000+200x+x^{2}\right)\times 1=204x$$

Whakamahia te āhuatanga tohatoha hei whakarea te 10000+200x+x^{2} ki te 1.

$$10000+200x+x^{2}=204x$$

Tangohia te 204x mai i ngā taha e rua.

$$10000+200x+x^{2}-204x=0$$

Pahekotia te 200x me -204x, ka -4x.

$$10000-4x+x^{2}=0$$

Tangohia te 10000 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$-4x+x^{2}=-10000$$

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.

$$x^{2}-4x=-10000$$

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

$$x^{2}-4x+\left(-2\right)^{2}=-10000+\left(-2\right)^{2}$$

Pūrua -2.

$$x^{2}-4x+4=-10000+4$$

Tāpiri -10000 ki te 4.

$$x^{2}-4x+4=-9996$$

Tauwehea x^{2}-4x+4. 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}.

$$\left(x-2\right)^{2}=-9996$$

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

$$\sqrt{\left(x-2\right)^{2}}=\sqrt{-9996}$$

Whakarūnātia.

$$x-2=14\sqrt{51}i x-2=-14\sqrt{51}i$$

Me tāpiri 2 ki ngā taha e rua o te whārite.

$$x=2+14\sqrt{51}i x=-14\sqrt{51}i+2$$