Whakaoti i te sqrt{15+x^2}-sqrt{19-x^2}=2

$$\sqrt { 15 + x ^ { 2 } } - \sqrt { 19 - x ^ { 2 } } = 2$$

Whakaoti mō x

$$x=\sqrt{10}\approx 3.16227766$$

$$x=-\sqrt{10}\approx -3.16227766$$

Ngā Upane Otinga

Me tango -\sqrt{19-x^{2}} mai i ngā taha e rua o te whārite.

$$\sqrt{15+x^{2}}=2+\sqrt{19-x^{2}}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{15+x^{2}}\right)^{2}=\left(2+\sqrt{19-x^{2}}\right)^{2}$$

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

$$15+x^{2}=\left(2+\sqrt{19-x^{2}}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(2+\sqrt{19-x^{2}}\right)^{2}.

$$15+x^{2}=4+4\sqrt{19-x^{2}}+\left(\sqrt{19-x^{2}}\right)^{2}$$

Tātaihia te \sqrt{19-x^{2}} mā te pū o 2, kia riro ko 19-x^{2}.

$$15+x^{2}=4+4\sqrt{19-x^{2}}+19-x^{2}$$

Tāpirihia te 4 ki te 19, ka 23.

$$15+x^{2}=23+4\sqrt{19-x^{2}}-x^{2}$$

Me tango 23-x^{2} mai i ngā taha e rua o te whārite.

$$15+x^{2}-\left(23-x^{2}\right)=4\sqrt{19-x^{2}}$$

Hei kimi i te tauaro o 23-x^{2}, kimihia te tauaro o ia taurangi.

$$15+x^{2}-23+x^{2}=4\sqrt{19-x^{2}}$$

Tangohia te 23 i te 15, ka -8.

$$-8+x^{2}+x^{2}=4\sqrt{19-x^{2}}$$

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

$$-8+2x^{2}=4\sqrt{19-x^{2}}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(-8+2x^{2}\right)^{2}=\left(4\sqrt{19-x^{2}}\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(-8+2x^{2}\right)^{2}.

$$64-32x^{2}+4\left(x^{2}\right)^{2}=\left(4\sqrt{19-x^{2}}\right)^{2}$$

Hei hiki pū ki tētahi pū anō, me whakarea ngā taupū. Me whakarea te 2 me te 2 kia riro ai te 4.

$$64-32x^{2}+4x^{4}=\left(4\sqrt{19-x^{2}}\right)^{2}$$

Whakarohaina te \left(4\sqrt{19-x^{2}}\right)^{2}.

$$64-32x^{2}+4x^{4}=4^{2}\left(\sqrt{19-x^{2}}\right)^{2}$$

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

$$64-32x^{2}+4x^{4}=16\left(\sqrt{19-x^{2}}\right)^{2}$$

Tātaihia te \sqrt{19-x^{2}} mā te pū o 2, kia riro ko 19-x^{2}.

$$64-32x^{2}+4x^{4}=16\left(19-x^{2}\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te 19-x^{2}.

$$64-32x^{2}+4x^{4}=304-16x^{2}$$

Tangohia te 304 mai i ngā taha e rua.

$$64-32x^{2}+4x^{4}-304=-16x^{2}$$

Tangohia te 304 i te 64, ka -240.

$$-240-32x^{2}+4x^{4}=-16x^{2}$$

Me tāpiri te 16x^{2} ki ngā taha e rua.

$$-240-32x^{2}+4x^{4}+16x^{2}=0$$

Pahekotia te -32x^{2} me 16x^{2}, ka -16x^{2}.

$$-240-16x^{2}+4x^{4}=0$$

Whakakapia te t mō te x^{2}.

$$4t^{2}-16t-240=0$$

Ka taea ngā whārite katoa o te momo ax^{2}+bx+c=0 te whakaoti mā te ture pūrua: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. Whakakapia te 4 mō te a, te -16 mō te b, me te -240 mō te c i te ture pūrua.

$$t=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 4\left(-240\right)}}{2\times 4}$$

Mahia ngā tātaitai.

$$t=\frac{16±64}{8}$$

Whakaotia te whārite t=\frac{16±64}{8} ina he tōrunga te ±, ina he tōraro te ±.

$$t=10 t=-6$$

I te mea ko x=t^{2}, ka riro ngā otinga mā te arotake i te x=±\sqrt{t} mō t tōrunga.

$$x=\sqrt{10} x=-\sqrt{10}$$

Whakakapia te \sqrt{10} mō te x i te whārite \sqrt{15+x^{2}}-\sqrt{19-x^{2}}=2.

$$\sqrt{15+\left(\sqrt{10}\right)^{2}}-\sqrt{19-\left(\sqrt{10}\right)^{2}}=2$$

Whakarūnātia. Ko te uara x=\sqrt{10} kua ngata te whārite.

$$2=2$$

Whakakapia te -\sqrt{10} mō te x i te whārite \sqrt{15+x^{2}}-\sqrt{19-x^{2}}=2.

$$\sqrt{15+\left(-\sqrt{10}\right)^{2}}-\sqrt{19-\left(-\sqrt{10}\right)^{2}}=2$$

Whakarūnātia. Ko te uara x=-\sqrt{10} kua ngata te whārite.

$$2=2$$

Rārangihia ngā rongoā katoa o \sqrt{x^{2}+15}=\sqrt{19-x^{2}}+2.

$$x=\sqrt{10} x=-\sqrt{10}$$