Whakaoti i te sqrt{x-1}-2=2sqrt{x+3}

$$\sqrt { x - 1 } - 2 = 2 \sqrt { x + 3 }$$

Whakaoti mō x (complex solution)

$$x=\frac{-16\sqrt{2}i-19}{9}\approx -2.111111111-2.514157444i$$

Ngā Upane Otinga

Pūruatia ngā taha e rua o te whārite.

$$\left(\sqrt{x-1}-2\right)^{2}=\left(2\sqrt{x+3}\right)^{2}$$

Whakamahia te ture huarua \left(a-b\right)^{2}=a^{2}-2ab+b^{2} hei whakaroha \left(\sqrt{x-1}-2\right)^{2}.

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

Tātaihia te \sqrt{x-1} mā te pū o 2, kia riro ko x-1.

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

Tāpirihia te -1 ki te 4, ka 3.

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

Whakarohaina te \left(2\sqrt{x+3}\right)^{2}.

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

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

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

Tātaihia te \sqrt{x+3} mā te pū o 2, kia riro ko x+3.

$$x+3-4\sqrt{x-1}=4\left(x+3\right)$$

Whakamahia te āhuatanga tohatoha hei whakarea te 4 ki te x+3.

$$x+3-4\sqrt{x-1}=4x+12$$

Me tango x+3 mai i ngā taha e rua o te whārite.

$$-4\sqrt{x-1}=4x+12-\left(x+3\right)$$

Hei kimi i te tauaro o x+3, kimihia te tauaro o ia taurangi.

$$-4\sqrt{x-1}=4x+12-x-3$$

Pahekotia te 4x me -x, ka 3x.

$$-4\sqrt{x-1}=3x+12-3$$

Tangohia te 3 i te 12, ka 9.

$$-4\sqrt{x-1}=3x+9$$

Pūruatia ngā taha e rua o te whārite.

$$\left(-4\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

Whakarohaina te \left(-4\sqrt{x-1}\right)^{2}.

$$\left(-4\right)^{2}\left(\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

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

$$16\left(\sqrt{x-1}\right)^{2}=\left(3x+9\right)^{2}$$

Tātaihia te \sqrt{x-1} mā te pū o 2, kia riro ko x-1.

$$16\left(x-1\right)=\left(3x+9\right)^{2}$$

Whakamahia te āhuatanga tohatoha hei whakarea te 16 ki te x-1.

$$16x-16=\left(3x+9\right)^{2}$$

Whakamahia te ture huarua \left(a+b\right)^{2}=a^{2}+2ab+b^{2} hei whakaroha \left(3x+9\right)^{2}.

$$16x-16=9x^{2}+54x+81$$

Tangohia te 9x^{2} mai i ngā taha e rua.

$$16x-16-9x^{2}=54x+81$$

Tangohia te 54x mai i ngā taha e rua.

$$16x-16-9x^{2}-54x=81$$

Pahekotia te 16x me -54x, ka -38x.

$$-38x-16-9x^{2}=81$$

Tangohia te 81 mai i ngā taha e rua.

$$-38x-16-9x^{2}-81=0$$

Tangohia te 81 i te -16, ka -97.

$$-38x-97-9x^{2}=0$$

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.

$$-9x^{2}-38x-97=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -9 mō a, -38 mō b, me -97 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-38\right)±\sqrt{\left(-38\right)^{2}-4\left(-9\right)\left(-97\right)}}{2\left(-9\right)}$$

Pūrua -38.

$$x=\frac{-\left(-38\right)±\sqrt{1444-4\left(-9\right)\left(-97\right)}}{2\left(-9\right)}$$

Whakareatia -4 ki te -9.

$$x=\frac{-\left(-38\right)±\sqrt{1444+36\left(-97\right)}}{2\left(-9\right)}$$

Whakareatia 36 ki te -97.

$$x=\frac{-\left(-38\right)±\sqrt{1444-3492}}{2\left(-9\right)}$$

Tāpiri 1444 ki te -3492.

$$x=\frac{-\left(-38\right)±\sqrt{-2048}}{2\left(-9\right)}$$

Tuhia te pūtakerua o te -2048.

$$x=\frac{-\left(-38\right)±32\sqrt{2}i}{2\left(-9\right)}$$

Ko te tauaro o -38 ko 38.

$$x=\frac{38±32\sqrt{2}i}{2\left(-9\right)}$$

Whakareatia 2 ki te -9.

$$x=\frac{38±32\sqrt{2}i}{-18}$$

Nā, me whakaoti te whārite x=\frac{38±32\sqrt{2}i}{-18} ina he tāpiri te ±. Tāpiri 38 ki te 32i\sqrt{2}.

$$x=\frac{38+32\sqrt{2}i}{-18}$$

Whakawehe 38+32i\sqrt{2} ki te -18.

$$x=\frac{-16\sqrt{2}i-19}{9}$$

Nā, me whakaoti te whārite x=\frac{38±32\sqrt{2}i}{-18} ina he tango te ±. Tango 32i\sqrt{2} mai i 38.

$$x=\frac{-32\sqrt{2}i+38}{-18}$$

Whakawehe 38-32i\sqrt{2} ki te -18.

$$x=\frac{-19+16\sqrt{2}i}{9}$$

Kua oti te whārite te whakatau.

$$x=\frac{-16\sqrt{2}i-19}{9} x=\frac{-19+16\sqrt{2}i}{9}$$

Whakakapia te \frac{-16\sqrt{2}i-19}{9} mō te x i te whārite \sqrt{x-1}-2=2\sqrt{x+3}.

$$\sqrt{\frac{-16\sqrt{2}i-19}{9}-1}-2=2\sqrt{\frac{-16\sqrt{2}i-19}{9}+3}$$

Whakarūnātia. Ko te uara x=\frac{-16\sqrt{2}i-19}{9} kua ngata te whārite.

$$-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

Whakakapia te \frac{-19+16\sqrt{2}i}{9} mō te x i te whārite \sqrt{x-1}-2=2\sqrt{x+3}.

$$\sqrt{\frac{-19+16\sqrt{2}i}{9}-1}-2=2\sqrt{\frac{-19+16\sqrt{2}i}{9}+3}$$

Whakarūnātia. Ko te uara x=\frac{-19+16\sqrt{2}i}{9} kāore e ngata ana ki te whārite.

$$-\frac{4}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

Whakakapia te \frac{-16\sqrt{2}i-19}{9} mō te x i te whārite \sqrt{x-1}-2=2\sqrt{x+3}.

$$\sqrt{\frac{-16\sqrt{2}i-19}{9}-1}-2=2\sqrt{\frac{-16\sqrt{2}i-19}{9}+3}$$

Whakarūnātia. Ko te uara x=\frac{-16\sqrt{2}i-19}{9} kua ngata te whārite.

$$-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}=-\frac{8}{3}+\frac{4}{3}i\times 2^{\frac{1}{2}}$$

Ko te whārite \sqrt{x-1}-2=2\sqrt{x+3} he rongoā ahurei.

$$x=\frac{-16\sqrt{2}i-19}{9}$$