Whakaoti i te x^2=-x^2+4x+1

$$x ^ { 2 } = - x ^ { 2 } + 4 x + 1$$

Whakaoti mō x

$$x=\frac{\sqrt{6}}{2}+1\approx 2.224744871$$

$$x=-\frac{\sqrt{6}}{2}+1\approx -0.224744871$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te x^{2} ki ngā taha e rua.

$$x^{2}+x^{2}=4x+1$$

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

$$2x^{2}=4x+1$$

Tangohia te 4x mai i ngā taha e rua.

$$2x^{2}-4x=1$$

Tangohia te 1 mai i ngā taha e rua.

$$2x^{2}-4x-1=0$$

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

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

Pūrua -4.

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

Whakareatia -4 ki te 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-1\right)}}{2\times 2}$$

Whakareatia -8 ki te -1.

$$x=\frac{-\left(-4\right)±\sqrt{16+8}}{2\times 2}$$

Tāpiri 16 ki te 8.

$$x=\frac{-\left(-4\right)±\sqrt{24}}{2\times 2}$$

Tuhia te pūtakerua o te 24.

$$x=\frac{-\left(-4\right)±2\sqrt{6}}{2\times 2}$$

Ko te tauaro o -4 ko 4.

$$x=\frac{4±2\sqrt{6}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$x=\frac{4±2\sqrt{6}}{4}$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{6}}{4} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{6}.

$$x=\frac{2\sqrt{6}+4}{4}$$

Whakawehe 4+2\sqrt{6} ki te 4.

$$x=\frac{\sqrt{6}}{2}+1$$

Nā, me whakaoti te whārite x=\frac{4±2\sqrt{6}}{4} ina he tango te ±. Tango 2\sqrt{6} mai i 4.

$$x=\frac{4-2\sqrt{6}}{4}$$

Whakawehe 4-2\sqrt{6} ki te 4.

$$x=-\frac{\sqrt{6}}{2}+1$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{6}}{2}+1 x=-\frac{\sqrt{6}}{2}+1$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te x^{2} ki ngā taha e rua.

$$x^{2}+x^{2}=4x+1$$

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

$$2x^{2}=4x+1$$

Tangohia te 4x mai i ngā taha e rua.

$$2x^{2}-4x=1$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2x^{2}-4x}{2}=\frac{1}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

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

Whakawehe -4 ki te 2.

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

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.

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

Tāpiri \frac{1}{2} ki te 1.

$$x^{2}-2x+1=\frac{3}{2}$$

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}.

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

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

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

Whakarūnātia.

$$x-1=\frac{\sqrt{6}}{2} x-1=-\frac{\sqrt{6}}{2}$$

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

$$x=\frac{\sqrt{6}}{2}+1 x=-\frac{\sqrt{6}}{2}+1$$