Whakaoti i te 15000=10000left(1+xright)^2

$$15000 = 10000 { \left(1+x \right) }^{ 2 }$$

Whakaoti mō x

$$x=\frac{\sqrt{6}}{2}-1\approx 0.224744871$$

$$x=-\frac{\sqrt{6}}{2}-1\approx -2.224744871$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawehea ngā taha e rua ki te 10000.

$$\frac{15000}{10000}=\left(1+x\right)^{2}$$

Whakahekea te hautanga \frac{15000}{10000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5000.

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$\frac{3}{2}=1+2x+x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1+2x+x^{2}=\frac{3}{2}$$

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

$$1+2x+x^{2}-\frac{3}{2}=0$$

Tangohia te \frac{3}{2} i te 1, ka -\frac{1}{2}.

$$-\frac{1}{2}+2x+x^{2}=0$$

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.

$$x^{2}+2x-\frac{1}{2}=0$$

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

$$x=\frac{-2±\sqrt{2^{2}-4\left(-\frac{1}{2}\right)}}{2}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\left(-\frac{1}{2}\right)}}{2}$$

Whakareatia -4 ki te -\frac{1}{2}.

$$x=\frac{-2±\sqrt{4+2}}{2}$$

Tāpiri 4 ki te 2.

$$x=\frac{-2±\sqrt{6}}{2}$$

Nā, me whakaoti te whārite x=\frac{-2±\sqrt{6}}{2} ina he tāpiri te ±. Tāpiri -2 ki te \sqrt{6}.

$$x=\frac{\sqrt{6}-2}{2}$$

Whakawehe -2+\sqrt{6} ki te 2.

$$x=\frac{\sqrt{6}}{2}-1$$

Nā, me whakaoti te whārite x=\frac{-2±\sqrt{6}}{2} ina he tango te ±. Tango \sqrt{6} mai i -2.

$$x=\frac{-\sqrt{6}-2}{2}$$

Whakawehe -2-\sqrt{6} ki te 2.

$$x=-\frac{\sqrt{6}}{2}-1$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{6}}{2}-1 x=-\frac{\sqrt{6}}{2}-1$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakawehea ngā taha e rua ki te 10000.

$$\frac{15000}{10000}=\left(1+x\right)^{2}$$

Whakahekea te hautanga \frac{15000}{10000} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5000.

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

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(1+x\right)^{2}.

$$\frac{3}{2}=1+2x+x^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$1+2x+x^{2}=\frac{3}{2}$$

Tangohia te 1 mai i ngā taha e rua.

$$2x+x^{2}=\frac{3}{2}-1$$

Tangohia te 1 i te \frac{3}{2}, ka \frac{1}{2}.

$$2x+x^{2}=\frac{1}{2}$$

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.

$$x^{2}+2x=\frac{1}{2}$$

Whakawehea te 2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 1. Nā, tāpiria te pūrua o te 1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+2x+1^{2}=\frac{1}{2}+1^{2}$$

Pūrua 1.

$$x^{2}+2x+1=\frac{1}{2}+1$$

Tāpiri \frac{1}{2} ki te 1.

$$x^{2}+2x+1=\frac{3}{2}$$

Tauwehea x^{2}+2x+1. 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}.

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

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

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

Whakarūnātia.

$$x+1=\frac{\sqrt{6}}{2} x+1=-\frac{\sqrt{6}}{2}$$

Me tango 1 mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{6}}{2}-1 x=-\frac{\sqrt{6}}{2}-1$$