Whakaoti i te sqrt{1/2+1/4+1/8+1/16+frac{1}{2}x}=x

$$\sqrt{ \frac{ 1 }{ 2 } + \frac{ 1 }{ 4 } + \frac{ 1 }{ 8 } + \frac{ 1 }{ 16 } + \frac{ 1 }{ 2 } x } = x$$

Whakaoti mō x

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

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

Ko te maha noa iti rawa atu o 2 me 4 ko 4. Me tahuri \frac{1}{2} me \frac{1}{4} ki te hautau me te tautūnga 4.

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

Tā te mea he rite te tauraro o \frac{2}{4} me \frac{1}{4}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

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

Tāpirihia te 2 ki te 1, ka 3.

$$\left(\sqrt{\frac{3}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

Ko te maha noa iti rawa atu o 4 me 8 ko 8. Me tahuri \frac{3}{4} me \frac{1}{8} ki te hautau me te tautūnga 8.

$$\left(\sqrt{\frac{6}{8}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

Tā te mea he rite te tauraro o \frac{6}{8} me \frac{1}{8}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\left(\sqrt{\frac{6+1}{8}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

Tāpirihia te 6 ki te 1, ka 7.

$$\left(\sqrt{\frac{7}{8}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

Ko te maha noa iti rawa atu o 8 me 16 ko 16. Me tahuri \frac{7}{8} me \frac{1}{16} ki te hautau me te tautūnga 16.

$$\left(\sqrt{\frac{14}{16}+\frac{1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

Tā te mea he rite te tauraro o \frac{14}{16} me \frac{1}{16}, me tāpiri rāua mā te tāpiri i ō raua taurunga.

$$\left(\sqrt{\frac{14+1}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

Tāpirihia te 14 ki te 1, ka 15.

$$\left(\sqrt{\frac{15}{16}+\frac{1}{2}x}\right)^{2}=x^{2}$$

Tātaihia te \sqrt{\frac{15}{16}+\frac{1}{2}x} mā te pū o 2, kia riro ko \frac{15}{16}+\frac{1}{2}x.

$$\frac{15}{16}+\frac{1}{2}x=x^{2}$$

Tangohia te x^{2} mai i ngā taha e rua.

$$\frac{15}{16}+\frac{1}{2}x-x^{2}=0$$

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.

$$-x^{2}+\frac{1}{2}x+\frac{15}{16}=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -1 mō a, \frac{1}{2} mō b, me \frac{15}{16} mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\frac{1}{2}±\sqrt{\left(\frac{1}{2}\right)^{2}-4\left(-1\right)\times \frac{15}{16}}}{2\left(-1\right)}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}-4\left(-1\right)\times \frac{15}{16}}}{2\left(-1\right)}$$

Whakareatia -4 ki te -1.

$$x=\frac{-\frac{1}{2}±\sqrt{\frac{1}{4}+4\times \frac{15}{16}}}{2\left(-1\right)}$$

Whakareatia 4 ki te \frac{15}{16}.

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

Tāpiri \frac{1}{4} ki te \frac{15}{4} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x=\frac{-\frac{1}{2}±\sqrt{4}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 4.

$$x=\frac{-\frac{1}{2}±2}{2\left(-1\right)}$$

Whakareatia 2 ki te -1.

$$x=\frac{-\frac{1}{2}±2}{-2}$$

Nā, me whakaoti te whārite x=\frac{-\frac{1}{2}±2}{-2} ina he tāpiri te ±. Tāpiri -\frac{1}{2} ki te 2.

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

Whakawehe \frac{3}{2} ki te -2.

$$x=-\frac{3}{4}$$

Nā, me whakaoti te whārite x=\frac{-\frac{1}{2}±2}{-2} ina he tango te ±. Tango 2 mai i -\frac{1}{2}.

$$x=-\frac{\frac{5}{2}}{-2}$$

Whakawehe -\frac{5}{2} ki te -2.

$$x=\frac{5}{4}$$

Kua oti te whārite te whakatau.

$$x=-\frac{3}{4} x=\frac{5}{4}$$

Whakakapia te -\frac{3}{4} mō te x i te whārite \sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}x}=x.

$$\sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}\left(-\frac{3}{4}\right)}=-\frac{3}{4}$$

Whakarūnātia. Ko te uara x=-\frac{3}{4} kāore e ngata ana ki te whārite nā te mea e rerekē ngā tohu o te taha maui me te taha katau.

$$\frac{3}{4}=-\frac{3}{4}$$

Whakakapia te \frac{5}{4} mō te x i te whārite \sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}x}=x.

$$\sqrt{\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\frac{1}{16}+\frac{1}{2}\times \frac{5}{4}}=\frac{5}{4}$$

Whakarūnātia. Ko te uara x=\frac{5}{4} kua ngata te whārite.

$$\frac{5}{4}=\frac{5}{4}$$

Ko te whārite \sqrt{\frac{x}{2}+\frac{15}{16}}=x he rongoā ahurei.

$$x=\frac{5}{4}$$