Whakaoti i te x+1/x^2+1-1/2=0

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

Whakaoti mō x

$$x=\sqrt{2}+1\approx 2.414213562$$

$$x=1-\sqrt{2}\approx -0.414213562$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me whakarea ngā taha e rua o te whārite ki te 2\left(x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+1,2.

$$2\left(x+1\right)+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

Whakareatia te 2 ki te -\frac{1}{2}, ka -1.

$$2x+2-\left(x^{2}+1\right)=0$$

Hei kimi i te tauaro o x^{2}+1, kimihia te tauaro o ia taurangi.

$$2x+2-x^{2}-1=0$$

Tangohia te 1 i te 2, ka 1.

$$2x+1-x^{2}=0$$

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.

$$-x^{2}+2x+1=0$$

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

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

Pūrua 2.

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

Whakareatia -4 ki te -1.

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

Tāpiri 4 ki te 4.

$$x=\frac{-2±\sqrt{8}}{2\left(-1\right)}$$

Tuhia te pūtakerua o te 8.

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

Whakareatia 2 ki te -1.

$$x=\frac{-2±2\sqrt{2}}{-2}$$

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

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

Whakawehe -2+2\sqrt{2} ki te -2.

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

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

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

Whakawehe -2-2\sqrt{2} ki te -2.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Me whakarea ngā taha e rua o te whārite ki te 2\left(x^{2}+1\right), arā, te tauraro pātahi he tino iti rawa te kitea o x^{2}+1,2.

$$2\left(x+1\right)+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

Whakamahia te āhuatanga tohatoha hei whakarea te 2 ki te x+1.

$$2x+2+2\left(x^{2}+1\right)\left(-\frac{1}{2}\right)=0$$

Whakareatia te 2 ki te -\frac{1}{2}, ka -1.

$$2x+2-\left(x^{2}+1\right)=0$$

Hei kimi i te tauaro o x^{2}+1, kimihia te tauaro o ia taurangi.

$$2x+2-x^{2}-1=0$$

Tangohia te 1 i te 2, ka 1.

$$2x+1-x^{2}=0$$

Tangohia te 1 mai i ngā taha e rua. Ko te tau i tango i te kore ka hua ko tōna korenga.

$$2x-x^{2}=-1$$

Ko ngā whārite pūrua pēnei i tēnei nā ka taea te whakaoti mā te whakaoti i te pūrua. Hei whakaoti i te pūrua, ko te whārite me mātua tuhi ki te āhua x^{2}+bx=c.

$$-x^{2}+2x=-1$$

Whakawehea ngā taha e rua ki te -1.

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

Mā te whakawehe ki te -1 ka wetekia te whakareanga ki te -1.

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

Whakawehe 2 ki te -1.

$$x^{2}-2x=-\frac{1}{-1}$$

Whakawehe -1 ki te -1.

$$x^{2}-2x=1$$

Whakawehea te -2, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -1. Nā, tāpiria te pūrua o te -1 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-2x+1=1+1$$

Tāpiri 1 ki te 1.

$$x^{2}-2x+1=2$$

Tauwehea x^{2}-2x+1. Ko te tikanga pūnoa, ina ko x^{2}+bx+c he pūrua tika pūrua tika pū, ka taea taua mea te tauwehea i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

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

Tuhia te pūtakerua o ngā taha e rua o te whārite.

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

Whakarūnātia.

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

Me tāpiri 1 ki ngā taha e rua o te whārite.

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