Whakaoti i te 10^2+x^2=8^2-(12-x)^2

$$10 ^ { 2 } + x ^ { 2 } = 8 ^ { 2 } - ( 12 - x ) ^ { 2 }$$

Whakaoti mō x (complex solution)

$$x=6+3\sqrt{6}i\approx 6+7.348469228i$$

$$x=-3\sqrt{6}i+6\approx 6-7.348469228i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 10 mā te pū o 2, kia riro ko 100.

$$100+x^{2}=8^{2}-\left(12-x\right)^{2}$$

Tātaihia te 8 mā te pū o 2, kia riro ko 64.

$$100+x^{2}=64-\left(12-x\right)^{2}$$

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

$$100+x^{2}=64-\left(144-24x+x^{2}\right)$$

Hei kimi i te tauaro o 144-24x+x^{2}, kimihia te tauaro o ia taurangi.

$$100+x^{2}=64-144+24x-x^{2}$$

Tangohia te 144 i te 64, ka -80.

$$100+x^{2}=-80+24x-x^{2}$$

Tangohia te -80 mai i ngā taha e rua.

$$100+x^{2}-\left(-80\right)=24x-x^{2}$$

Ko te tauaro o -80 ko 80.

$$100+x^{2}+80=24x-x^{2}$$

Tangohia te 24x mai i ngā taha e rua.

$$100+x^{2}+80-24x=-x^{2}$$

Tāpirihia te 100 ki te 80, ka 180.

$$180+x^{2}-24x=-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$180+x^{2}-24x+x^{2}=0$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$180+2x^{2}-24x=0$$

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.

$$2x^{2}-24x+180=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -24 mō b, me 180 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 2\times 180}}{2\times 2}$$

Pūrua -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 2\times 180}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-24\right)±\sqrt{576-8\times 180}}{2\times 2}$$

Whakareatia -8 ki te 180.

$$x=\frac{-\left(-24\right)±\sqrt{576-1440}}{2\times 2}$$

Tāpiri 576 ki te -1440.

$$x=\frac{-\left(-24\right)±\sqrt{-864}}{2\times 2}$$

Tuhia te pūtakerua o te -864.

$$x=\frac{-\left(-24\right)±12\sqrt{6}i}{2\times 2}$$

Ko te tauaro o -24 ko 24.

$$x=\frac{24±12\sqrt{6}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{24±12\sqrt{6}i}{4}$$

Nā, me whakaoti te whārite x=\frac{24±12\sqrt{6}i}{4} ina he tāpiri te ±. Tāpiri 24 ki te 12i\sqrt{6}.

$$x=\frac{24+12\sqrt{6}i}{4}$$

Whakawehe 24+12i\sqrt{6} ki te 4.

$$x=6+3\sqrt{6}i$$

Nā, me whakaoti te whārite x=\frac{24±12\sqrt{6}i}{4} ina he tango te ±. Tango 12i\sqrt{6} mai i 24.

$$x=\frac{-12\sqrt{6}i+24}{4}$$

Whakawehe 24-12i\sqrt{6} ki te 4.

$$x=-3\sqrt{6}i+6$$

Kua oti te whārite te whakatau.

$$x=6+3\sqrt{6}i x=-3\sqrt{6}i+6$$

Ngā Upane mō te Whakaoti i te Pūrua

Tātaihia te 10 mā te pū o 2, kia riro ko 100.

$$100+x^{2}=8^{2}-\left(12-x\right)^{2}$$

Tātaihia te 8 mā te pū o 2, kia riro ko 64.

$$100+x^{2}=64-\left(12-x\right)^{2}$$

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

$$100+x^{2}=64-\left(144-24x+x^{2}\right)$$

Hei kimi i te tauaro o 144-24x+x^{2}, kimihia te tauaro o ia taurangi.

$$100+x^{2}=64-144+24x-x^{2}$$

Tangohia te 144 i te 64, ka -80.

$$100+x^{2}=-80+24x-x^{2}$$

Tangohia te 24x mai i ngā taha e rua.

$$100+x^{2}-24x=-80-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$100+x^{2}-24x+x^{2}=-80$$

Pahekotia te x^{2} me x^{2}, ka 2x^{2}.

$$100+2x^{2}-24x=-80$$

Tangohia te 100 mai i ngā taha e rua.

$$2x^{2}-24x=-80-100$$

Tangohia te 100 i te -80, ka -180.

$$2x^{2}-24x=-180$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-24x}{2}=-\frac{180}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

$$x^{2}+\left(-\frac{24}{2}\right)x=-\frac{180}{2}$$

Whakawehe -24 ki te 2.

$$x^{2}-12x=-\frac{180}{2}$$

Whakawehe -180 ki te 2.

$$x^{2}-12x=-90$$

Whakawehea te -12, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -6. Nā, tāpiria te pūrua o te -6 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-12x+\left(-6\right)^{2}=-90+\left(-6\right)^{2}$$

Pūrua -6.

$$x^{2}-12x+36=-90+36$$

Tāpiri -90 ki te 36.

$$x^{2}-12x+36=-54$$

Tauwehea x^{2}-12x+36. 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}.

$$\left(x-6\right)^{2}=-54$$

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

$$\sqrt{\left(x-6\right)^{2}}=\sqrt{-54}$$

Whakarūnātia.

$$x-6=3\sqrt{6}i x-6=-3\sqrt{6}i$$

Me tāpiri 6 ki ngā taha e rua o te whārite.

$$x=6+3\sqrt{6}i x=-3\sqrt{6}i+6$$