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

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

Whakaoti mō x

$$x=\sqrt{2}+2\approx 3.414213562$$

$$x=2-\sqrt{2}\approx 0.585786438$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

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

Tangohia te 1 mai i ngā taha e rua.

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

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

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

Pūrua 2.

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

Whakareatia -4 ki te -\frac{1}{2}.

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

Whakareatia 2 ki te -1.

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

Tāpiri 4 ki te -2.

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

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

$$x=\frac{-2±\sqrt{2}}{-1}$$

Nā, me whakaoti te whārite x=\frac{-2±\sqrt{2}}{-1} ina he tāpiri te ±. Tāpiri -2 ki te \sqrt{2}.

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

Whakawehe -2+\sqrt{2} ki te -1.

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

Nā, me whakaoti te whārite x=\frac{-2±\sqrt{2}}{-1} ina he tango te ±. Tango \sqrt{2} mai i -2.

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

Whakawehe -2-\sqrt{2} ki te -1.

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

Kua oti te whārite te whakatau.

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

Ngā Upane mō te Whakaoti i te Pūrua

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

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

Me whakarea ngā taha e rua ki te -2.

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

Mā te whakawehe ki te -\frac{1}{2} ka wetekia te whakareanga ki te -\frac{1}{2}.

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

Whakawehe 2 ki te -\frac{1}{2} mā te whakarea 2 ki te tau huripoki o -\frac{1}{2}.

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

Whakawehe 1 ki te -\frac{1}{2} mā te whakarea 1 ki te tau huripoki o -\frac{1}{2}.

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

Whakawehea te -4, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -2. Nā, tāpiria te pūrua o te -2 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūrua -2.

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

Tāpiri -2 ki te 4.

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

Tauwehea x^{2}-4x+4. 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}.

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

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

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

Whakarūnātia.

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

Me tāpiri 2 ki ngā taha e rua o te whārite.

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