Whakaoti i te 6^2+(2*5+x)^2=4^2-(2*5-x)^2

$$6 ^ { 2 } + ( 2 \cdot 5 + x ) ^ { 2 } = 4 ^ { 2 } - ( 2 \cdot 5 - x ) ^ { 2 }$$

Whakaoti mō x (complex solution)

$$x=-\sqrt{110}i\approx -0-10.488088482i$$

$$x=\sqrt{110}i\approx 10.488088482i$$

Ngā Upane mā te Kimi Pūtakerua

Tātaihia te 6 mā te pū o 2, kia riro ko 36.

$$36+\left(2\times 5+x\right)^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Whakareatia te 2 ki te 5, ka 10.

$$36+\left(10+x\right)^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(10+x\right)^{2}.

$$36+100+20x+x^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Tāpirihia te 36 ki te 100, ka 136.

$$136+20x+x^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$136+20x+x^{2}=16-\left(2\times 5-x\right)^{2}$$

Whakareatia te 2 ki te 5, ka 10.

$$136+20x+x^{2}=16-\left(10-x\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(10-x\right)^{2}.

$$136+20x+x^{2}=16-\left(100-20x+x^{2}\right)$$

Hei kimi i te tauaro o 100-20x+x^{2}, kimihia te tauaro o ia taurangi.

$$136+20x+x^{2}=16-100+20x-x^{2}$$

Tangohia te 100 i te 16, ka -84.

$$136+20x+x^{2}=-84+20x-x^{2}$$

Tangohia te 20x mai i ngā taha e rua.

$$136+20x+x^{2}-20x=-84-x^{2}$$

Pahekotia te 20x me -20x, ka 0.

$$136+x^{2}=-84-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$136+x^{2}+x^{2}=-84$$

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

$$136+2x^{2}=-84$$

Tangohia te 136 mai i ngā taha e rua.

$$2x^{2}=-84-136$$

Tangohia te 136 i te -84, ka -220.

$$2x^{2}=-220$$

Whakawehea ngā taha e rua ki te 2.

$$x^{2}=\frac{-220}{2}$$

Whakawehea te -220 ki te 2, kia riro ko -110.

$$x^{2}=-110$$

Kua oti te whārite te whakatau.

$$x=\sqrt{110}i x=-\sqrt{110}i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tātaihia te 6 mā te pū o 2, kia riro ko 36.

$$36+\left(2\times 5+x\right)^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Whakareatia te 2 ki te 5, ka 10.

$$36+\left(10+x\right)^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(10+x\right)^{2}.

$$36+100+20x+x^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Tāpirihia te 36 ki te 100, ka 136.

$$136+20x+x^{2}=4^{2}-\left(2\times 5-x\right)^{2}$$

Tātaihia te 4 mā te pū o 2, kia riro ko 16.

$$136+20x+x^{2}=16-\left(2\times 5-x\right)^{2}$$

Whakareatia te 2 ki te 5, ka 10.

$$136+20x+x^{2}=16-\left(10-x\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(10-x\right)^{2}.

$$136+20x+x^{2}=16-\left(100-20x+x^{2}\right)$$

Hei kimi i te tauaro o 100-20x+x^{2}, kimihia te tauaro o ia taurangi.

$$136+20x+x^{2}=16-100+20x-x^{2}$$

Tangohia te 100 i te 16, ka -84.

$$136+20x+x^{2}=-84+20x-x^{2}$$

Tangohia te -84 mai i ngā taha e rua.

$$136+20x+x^{2}-\left(-84\right)=20x-x^{2}$$

Ko te tauaro o -84 ko 84.

$$136+20x+x^{2}+84=20x-x^{2}$$

Tangohia te 20x mai i ngā taha e rua.

$$136+20x+x^{2}+84-20x=-x^{2}$$

Tāpirihia te 136 ki te 84, ka 220.

$$220+20x+x^{2}-20x=-x^{2}$$

Pahekotia te 20x me -20x, ka 0.

$$220+x^{2}=-x^{2}$$

Me tāpiri te x^{2} ki ngā taha e rua.

$$220+x^{2}+x^{2}=0$$

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

$$220+2x^{2}=0$$

Ko ngā tikanga tātai pūrua pēnei i tēnei nā, me te kīanga tau x^{2} engari kāore he kīanga tau x, ka taea tonu te whakaoti mā te whakamahi i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, ina tuhia ki te tānga ngahuru: ax^{2}+bx+c=0.

$$2x^{2}+220=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, 0 mō b, me 220 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\times 220}}{2\times 2}$$

Pūrua 0.

$$x=\frac{0±\sqrt{-4\times 2\times 220}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$x=\frac{0±\sqrt{-8\times 220}}{2\times 2}$$

Whakareatia -8 ki te 220.

$$x=\frac{0±\sqrt{-1760}}{2\times 2}$$

Tuhia te pūtakerua o te -1760.

$$x=\frac{0±4\sqrt{110}i}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{0±4\sqrt{110}i}{4}$$

Nā, me whakaoti te whārite x=\frac{0±4\sqrt{110}i}{4} ina he tāpiri te ±.

$$x=\sqrt{110}i$$

Nā, me whakaoti te whārite x=\frac{0±4\sqrt{110}i}{4} ina he tango te ±.

$$x=-\sqrt{110}i$$

Kua oti te whārite te whakatau.

$$x=\sqrt{110}i x=-\sqrt{110}i$$