Whakaoti i te {l}{x={(4-sqrt{15})}}{y={(x^2+1/x^2)}}{z=y}{text{Solvefor}atext{where}}{a=z}

$$\left. \begin{array} { l } { x = {(4 - \sqrt{15})} }\\ { y = {(x ^ {2} + \frac{1}{x ^ {2}})} }\\ { z = y }\\ { \text{Solve for } a \text{ where} } \\ { a = z } \end{array} \right.$$

Whakaoti mō x, y, z, a

$$a=62$$

Ngā Upane Otinga

Whakaarohia te whārite tuarua. Me kōkuhu ngā uara tāupe mōhiotia ki te whārite.

$$y=\left(4-\sqrt{15}\right)^{2}+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(4-\sqrt{15}\right)^{2}.

$$y=16-8\sqrt{15}+\left(\sqrt{15}\right)^{2}+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

Ko te pūrua o \sqrt{15} ko 15.

$$y=16-8\sqrt{15}+15+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

Tāpirihia te 16 ki te 15, ka 31.

$$y=31-8\sqrt{15}+\frac{1}{\left(4-\sqrt{15}\right)^{2}}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(4-\sqrt{15}\right)^{2}.

$$y=31-8\sqrt{15}+\frac{1}{16-8\sqrt{15}+\left(\sqrt{15}\right)^{2}}$$

Ko te pūrua o \sqrt{15} ko 15.

$$y=31-8\sqrt{15}+\frac{1}{16-8\sqrt{15}+15}$$

Tāpirihia te 16 ki te 15, ka 31.

$$y=31-8\sqrt{15}+\frac{1}{31-8\sqrt{15}}$$

Whakangāwaritia te tauraro o \frac{1}{31-8\sqrt{15}} mā te whakarea i te taurunga me te tauraro ki te 31+8\sqrt{15}.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{\left(31-8\sqrt{15}\right)\left(31+8\sqrt{15}\right)}$$

Whakaarohia te \left(31-8\sqrt{15}\right)\left(31+8\sqrt{15}\right). Ka taea te whakareanga te panoni ki te rerekētanga o ngā pūrua mā te ture: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{31^{2}-\left(-8\sqrt{15}\right)^{2}}$$

Tātaihia te 31 mā te pū o 2, kia riro ko 961.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-\left(-8\sqrt{15}\right)^{2}}$$

Whakarohaina te \left(-8\sqrt{15}\right)^{2}.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-\left(-8\right)^{2}\left(\sqrt{15}\right)^{2}}$$

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

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-64\left(\sqrt{15}\right)^{2}}$$

Ko te pūrua o \sqrt{15} ko 15.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-64\times 15}$$

Whakareatia te 64 ki te 15, ka 960.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{961-960}$$

Tangohia te 960 i te 961, ka 1.

$$y=31-8\sqrt{15}+\frac{31+8\sqrt{15}}{1}$$

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

$$y=31-8\sqrt{15}+31+8\sqrt{15}$$

Tāpirihia te 31 ki te 31, ka 62.

$$y=62-8\sqrt{15}+8\sqrt{15}$$

Pahekotia te -8\sqrt{15} me 8\sqrt{15}, ka 0.

$$y=62$$

Whakaarohia te whārite tuatoru. Me kōkuhu ngā uara tāupe mōhiotia ki te whārite.

$$z=62$$

Whakaarohia te whārite tuawhā. Me kōkuhu ngā uara tāupe mōhiotia ki te whārite.

$$a=62$$

Kua oti te pūnaha te whakatau.

$$x=4-\sqrt{15} y=62 z=62 a=62$$