Whakaoti i te x^2+(14-x)^2=8^2

$$x ^ { 2 } + ( 14 - x ) ^ { 2 } = 8 ^ { 2 }$$

Whakaoti mō x (complex solution)

$$x=7+\sqrt{17}i\approx 7+4.123105626i$$

$$x=-\sqrt{17}i+7\approx 7-4.123105626i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(14-x\right)^{2}.

$$x^{2}+196-28x+x^{2}=8^{2}$$

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

$$2x^{2}+196-28x=8^{2}$$

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

$$2x^{2}+196-28x=64$$

Tangohia te 64 mai i ngā taha e rua.

$$2x^{2}+196-28x-64=0$$

Tangohia te 64 i te 196, ka 132.

$$2x^{2}+132-28x=0$$

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.

$$2x^{2}-28x+132=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -28 mō b, me 132 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\times 2\times 132}}{2\times 2}$$

Pūrua -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\times 2\times 132}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-28\right)±\sqrt{784-8\times 132}}{2\times 2}$$

Whakareatia -8 ki te 132.

$$x=\frac{-\left(-28\right)±\sqrt{784-1056}}{2\times 2}$$

Tāpiri 784 ki te -1056.

$$x=\frac{-\left(-28\right)±\sqrt{-272}}{2\times 2}$$

Tuhia te pūtakerua o te -272.

$$x=\frac{-\left(-28\right)±4\sqrt{17}i}{2\times 2}$$

Ko te tauaro o -28 ko 28.

$$x=\frac{28±4\sqrt{17}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{28±4\sqrt{17}i}{4}$$

Nā, me whakaoti te whārite x=\frac{28±4\sqrt{17}i}{4} ina he tāpiri te ±. Tāpiri 28 ki te 4i\sqrt{17}.

$$x=\frac{28+4\sqrt{17}i}{4}$$

Whakawehe 28+4i\sqrt{17} ki te 4.

$$x=7+\sqrt{17}i$$

Nā, me whakaoti te whārite x=\frac{28±4\sqrt{17}i}{4} ina he tango te ±. Tango 4i\sqrt{17} mai i 28.

$$x=\frac{-4\sqrt{17}i+28}{4}$$

Whakawehe 28-4i\sqrt{17} ki te 4.

$$x=-\sqrt{17}i+7$$

Kua oti te whārite te whakatau.

$$x=7+\sqrt{17}i x=-\sqrt{17}i+7$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(14-x\right)^{2}.

$$x^{2}+196-28x+x^{2}=8^{2}$$

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

$$2x^{2}+196-28x=8^{2}$$

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

$$2x^{2}+196-28x=64$$

Tangohia te 196 mai i ngā taha e rua.

$$2x^{2}-28x=64-196$$

Tangohia te 196 i te 64, ka -132.

$$2x^{2}-28x=-132$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-28x}{2}=-\frac{132}{2}$$

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

$$x^{2}+\left(-\frac{28}{2}\right)x=-\frac{132}{2}$$

Whakawehe -28 ki te 2.

$$x^{2}-14x=-\frac{132}{2}$$

Whakawehe -132 ki te 2.

$$x^{2}-14x=-66$$

Whakawehea te -14, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -7. Nā, tāpiria te pūrua o te -7 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-14x+\left(-7\right)^{2}=-66+\left(-7\right)^{2}$$

Pūrua -7.

$$x^{2}-14x+49=-66+49$$

Tāpiri -66 ki te 49.

$$x^{2}-14x+49=-17$$

Tauwehea x^{2}-14x+49. 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}.

$$\left(x-7\right)^{2}=-17$$

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

$$\sqrt{\left(x-7\right)^{2}}=\sqrt{-17}$$

Whakarūnātia.

$$x-7=\sqrt{17}i x-7=-\sqrt{17}i$$

Me tāpiri 7 ki ngā taha e rua o te whārite.

$$x=7+\sqrt{17}i x=-\sqrt{17}i+7$$