Whakaoti i te 2/1=(12+h)^2/12^2

$$\frac { 2 } { 1 } = \frac { ( 12 + h ) ^ { 2 } } { 12 ^ { 2 } }$$

Whakaoti mō h

$$h=12\sqrt{2}-12\approx 4.970562748$$

$$h=-12\sqrt{2}-12\approx -28.970562748$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Ka whakawehea he tau ki te tahi, hua ai ko ia anō.

$$2=\frac{\left(12+h\right)^{2}}{12^{2}}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(12+h\right)^{2}.

$$2=\frac{144+24h+h^{2}}{12^{2}}$$

Tātaihia te 12 mā te pū o 2, kia riro ko 144.

$$2=\frac{144+24h+h^{2}}{144}$$

Whakawehea ia wā o 144+24h+h^{2} ki te 144, kia riro ko 1+\frac{1}{6}h+\frac{1}{144}h^{2}.

$$2=1+\frac{1}{6}h+\frac{1}{144}h^{2}$$

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

$$1+\frac{1}{6}h+\frac{1}{144}h^{2}=2$$

Tangohia te 2 mai i ngā taha e rua.

$$1+\frac{1}{6}h+\frac{1}{144}h^{2}-2=0$$

Tangohia te 2 i te 1, ka -1.

$$-1+\frac{1}{6}h+\frac{1}{144}h^{2}=0$$

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.

$$\frac{1}{144}h^{2}+\frac{1}{6}h-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \frac{1}{144} mō a, \frac{1}{6} mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$h=\frac{-\frac{1}{6}±\sqrt{\left(\frac{1}{6}\right)^{2}-4\times \frac{1}{144}\left(-1\right)}}{2\times \frac{1}{144}}$$

Pūruatia \frac{1}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{36}-4\times \frac{1}{144}\left(-1\right)}}{2\times \frac{1}{144}}$$

Whakareatia -4 ki te \frac{1}{144}.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{36}-\frac{1}{36}\left(-1\right)}}{2\times \frac{1}{144}}$$

Whakareatia -\frac{1}{36} ki te -1.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1+1}{36}}}{2\times \frac{1}{144}}$$

Tāpiri \frac{1}{36} ki te \frac{1}{36} 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.

$$h=\frac{-\frac{1}{6}±\sqrt{\frac{1}{18}}}{2\times \frac{1}{144}}$$

Tuhia te pūtakerua o te \frac{1}{18}.

$$h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{2\times \frac{1}{144}}$$

Whakareatia 2 ki te \frac{1}{144}.

$$h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{\frac{1}{72}}$$

Nā, me whakaoti te whārite h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{\frac{1}{72}} ina he tāpiri te ±. Tāpiri -\frac{1}{6} ki te \frac{\sqrt{2}}{6}.

$$h=\frac{\sqrt{2}-1}{\frac{1}{72}\times 6}$$

Whakawehe \frac{-1+\sqrt{2}}{6} ki te \frac{1}{72} mā te whakarea \frac{-1+\sqrt{2}}{6} ki te tau huripoki o \frac{1}{72}.

$$h=12\sqrt{2}-12$$

Nā, me whakaoti te whārite h=\frac{-\frac{1}{6}±\frac{\sqrt{2}}{6}}{\frac{1}{72}} ina he tango te ±. Tango \frac{\sqrt{2}}{6} mai i -\frac{1}{6}.

$$h=\frac{-\sqrt{2}-1}{\frac{1}{72}\times 6}$$

Whakawehe \frac{-1-\sqrt{2}}{6} ki te \frac{1}{72} mā te whakarea \frac{-1-\sqrt{2}}{6} ki te tau huripoki o \frac{1}{72}.

$$h=-12\sqrt{2}-12$$

Kua oti te whārite te whakatau.

$$h=12\sqrt{2}-12 h=-12\sqrt{2}-12$$

Ngā Upane mō te Whakaoti i te Pūrua

Ka whakawehea he tau ki te tahi, hua ai ko ia anō.

$$2=\frac{\left(12+h\right)^{2}}{12^{2}}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(12+h\right)^{2}.

$$2=\frac{144+24h+h^{2}}{12^{2}}$$

Tātaihia te 12 mā te pū o 2, kia riro ko 144.

$$2=\frac{144+24h+h^{2}}{144}$$

Whakawehea ia wā o 144+24h+h^{2} ki te 144, kia riro ko 1+\frac{1}{6}h+\frac{1}{144}h^{2}.

$$2=1+\frac{1}{6}h+\frac{1}{144}h^{2}$$

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

$$1+\frac{1}{6}h+\frac{1}{144}h^{2}=2$$

Tangohia te 1 mai i ngā taha e rua.

$$\frac{1}{6}h+\frac{1}{144}h^{2}=2-1$$

Tangohia te 1 i te 2, ka 1.

$$\frac{1}{6}h+\frac{1}{144}h^{2}=1$$

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.

$$\frac{1}{144}h^{2}+\frac{1}{6}h=1$$

Me whakarea ngā taha e rua ki te 144.

$$\frac{\frac{1}{144}h^{2}+\frac{1}{6}h}{\frac{1}{144}}=\frac{1}{\frac{1}{144}}$$

Mā te whakawehe ki te \frac{1}{144} ka wetekia te whakareanga ki te \frac{1}{144}.

$$h^{2}+\frac{\frac{1}{6}}{\frac{1}{144}}h=\frac{1}{\frac{1}{144}}$$

Whakawehe \frac{1}{6} ki te \frac{1}{144} mā te whakarea \frac{1}{6} ki te tau huripoki o \frac{1}{144}.

$$h^{2}+24h=\frac{1}{\frac{1}{144}}$$

Whakawehe 1 ki te \frac{1}{144} mā te whakarea 1 ki te tau huripoki o \frac{1}{144}.

$$h^{2}+24h=144$$

Whakawehea te 24, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 12. Nā, tāpiria te pūrua o te 12 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$h^{2}+24h+12^{2}=144+12^{2}$$

Pūrua 12.

$$h^{2}+24h+144=144+144$$

Tāpiri 144 ki te 144.

$$h^{2}+24h+144=288$$

Tauwehea te h^{2}+24h+144. 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}.

$$\left(h+12\right)^{2}=288$$

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

$$\sqrt{\left(h+12\right)^{2}}=\sqrt{288}$$

Whakarūnātia.

$$h+12=12\sqrt{2} h+12=-12\sqrt{2}$$

Me tango 12 mai i ngā taha e rua o te whārite.

$$h=12\sqrt{2}-12 h=-12\sqrt{2}-12$$