Whakaoti i te x^2+11+sqrt{x^2+11}=42

$$x ^ { 2 } + 11 + \sqrt { x ^ { 2 } + 11 } = 42$$

Whakaoti mō x

$$x=-5$$

$$x=5$$

Ngā Upane Otinga

Me tango x^{2}+11 mai i ngā taha e rua o te whārite.

$$\sqrt{x^{2}+11}=42-\left(x^{2}+11\right)$$

Hei kimi i te tauaro o x^{2}+11, kimihia te tauaro o ia taurangi.

$$\sqrt{x^{2}+11}=42-x^{2}-11$$

Tangohia te 11 i te 42, ka 31.

$$\sqrt{x^{2}+11}=31-x^{2}$$

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{x^{2}+11}\right)^{2}=\left(31-x^{2}\right)^{2}$$

Tātaihia te \sqrt{x^{2}+11} mā te pū o 2, kia riro ko x^{2}+11.

$$x^{2}+11=\left(31-x^{2}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(31-x^{2}\right)^{2}.

$$x^{2}+11=961-62x^{2}+\left(x^{2}\right)^{2}$$

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

$$x^{2}+11=961-62x^{2}+x^{4}$$

Tangohia te 961 mai i ngā taha e rua.

$$x^{2}+11-961=-62x^{2}+x^{4}$$

Tangohia te 961 i te 11, ka -950.

$$x^{2}-950=-62x^{2}+x^{4}$$

Me tāpiri te 62x^{2} ki ngā taha e rua.

$$x^{2}-950+62x^{2}=x^{4}$$

Pahekotia te x^{2} me 62x^{2}, ka 63x^{2}.

$$63x^{2}-950=x^{4}$$

Tangohia te x^{4} mai i ngā taha e rua.

$$63x^{2}-950-x^{4}=0$$

Whakakapia te t mō te x^{2}.

$$-t^{2}+63t-950=0$$

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 -1 mō te a, te 63 mō te b, me te -950 mō te c i te ture pūrua.

$$t=\frac{-63±\sqrt{63^{2}-4\left(-1\right)\left(-950\right)}}{-2}$$

Mahia ngā tātaitai.

$$t=\frac{-63±13}{-2}$$

Whakaotia te whārite t=\frac{-63±13}{-2} ina he tōrunga te ±, ina he tōraro te ±.

$$t=25 t=38$$

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

$$x=5 x=-5 x=\sqrt{38} x=-\sqrt{38}$$

Whakakapia te 5 mō te x i te whārite x^{2}+11+\sqrt{x^{2}+11}=42.

$$5^{2}+11+\sqrt{5^{2}+11}=42$$

Whakarūnātia. Ko te uara x=5 kua ngata te whārite.

$$42=42$$

Whakakapia te -5 mō te x i te whārite x^{2}+11+\sqrt{x^{2}+11}=42.

$$\left(-5\right)^{2}+11+\sqrt{\left(-5\right)^{2}+11}=42$$

Whakarūnātia. Ko te uara x=-5 kua ngata te whārite.

$$42=42$$

Whakakapia te \sqrt{38} mō te x i te whārite x^{2}+11+\sqrt{x^{2}+11}=42.

$$\left(\sqrt{38}\right)^{2}+11+\sqrt{\left(\sqrt{38}\right)^{2}+11}=42$$

Whakarūnātia. Ko te uara x=\sqrt{38} kāore e ngata ana ki te whārite.

$$56=42$$

Whakakapia te -\sqrt{38} mō te x i te whārite x^{2}+11+\sqrt{x^{2}+11}=42.

$$\left(-\sqrt{38}\right)^{2}+11+\sqrt{\left(-\sqrt{38}\right)^{2}+11}=42$$

Whakarūnātia. Ko te uara x=-\sqrt{38} kāore e ngata ana ki te whārite.

$$56=42$$

Rārangihia ngā rongoā katoa o \sqrt{x^{2}+11}=31-x^{2}.

$$x=5 x=-5$$