Whakaoti i te x^2=(2+sqrt{5})^2+(2-sqrt{5})^2

$$x ^ { 2 } = ( 2 + \sqrt { 5 } ) ^ { 2 } + ( 2 - \sqrt { 5 } ) ^ { 2 }$$

Whakaoti mō x

$$x=3\sqrt{2}\approx 4.242640687$$

$$x=-3\sqrt{2}\approx -4.242640687$$

Ngā Upane mā te Kimi Pūtakerua

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2+\sqrt{5}\right)^{2}.

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

Ko te pūrua o \sqrt{5} ko 5.

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

Tāpirihia te 4 ki te 5, ka 9.

$$x^{2}=9+4\sqrt{5}+\left(2-\sqrt{5}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2-\sqrt{5}\right)^{2}.

$$x^{2}=9+4\sqrt{5}+4-4\sqrt{5}+\left(\sqrt{5}\right)^{2}$$

Ko te pūrua o \sqrt{5} ko 5.

$$x^{2}=9+4\sqrt{5}+4-4\sqrt{5}+5$$

Tāpirihia te 4 ki te 5, ka 9.

$$x^{2}=9+4\sqrt{5}+9-4\sqrt{5}$$

Tāpirihia te 9 ki te 9, ka 18.

$$x^{2}=18+4\sqrt{5}-4\sqrt{5}$$

Pahekotia te 4\sqrt{5} me -4\sqrt{5}, ka 0.

$$x^{2}=18$$

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

$$x=3\sqrt{2} x=-3\sqrt{2}$$

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(2+\sqrt{5}\right)^{2}.

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

Ko te pūrua o \sqrt{5} ko 5.

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

Tāpirihia te 4 ki te 5, ka 9.

$$x^{2}=9+4\sqrt{5}+\left(2-\sqrt{5}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(2-\sqrt{5}\right)^{2}.

$$x^{2}=9+4\sqrt{5}+4-4\sqrt{5}+\left(\sqrt{5}\right)^{2}$$

Ko te pūrua o \sqrt{5} ko 5.

$$x^{2}=9+4\sqrt{5}+4-4\sqrt{5}+5$$

Tāpirihia te 4 ki te 5, ka 9.

$$x^{2}=9+4\sqrt{5}+9-4\sqrt{5}$$

Tāpirihia te 9 ki te 9, ka 18.

$$x^{2}=18+4\sqrt{5}-4\sqrt{5}$$

Pahekotia te 4\sqrt{5} me -4\sqrt{5}, ka 0.

$$x^{2}=18$$

Tangohia te 18 mai i ngā taha e rua.

$$x^{2}-18=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 0 mō b, me -18 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-18\right)}}{2}$$

Pūrua 0.

$$x=\frac{0±\sqrt{-4\left(-18\right)}}{2}$$

Whakareatia -4 ki te -18.

$$x=\frac{0±\sqrt{72}}{2}$$

Tuhia te pūtakerua o te 72.

$$x=\frac{0±6\sqrt{2}}{2}$$

Nā, me whakaoti te whārite x=\frac{0±6\sqrt{2}}{2} ina he tāpiri te ±.

$$x=3\sqrt{2}$$

Nā, me whakaoti te whārite x=\frac{0±6\sqrt{2}}{2} ina he tango te ±.

$$x=-3\sqrt{2}$$

Kua oti te whārite te whakatau.

$$x=3\sqrt{2} x=-3\sqrt{2}$$