Whakaoti i te 4x=4x^2-8x+4

$$4 x = 4 x ^ { 2 } - 8 x + 4$$

Whakaoti mō x

$$x = \frac{\sqrt{5} + 3}{2} \approx 2.618033989$$

$$x=\frac{3-\sqrt{5}}{2}\approx 0.381966011$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tangohia te 4x^{2} mai i ngā taha e rua.

$$4x-4x^{2}=-8x+4$$

Me tāpiri te 8x ki ngā taha e rua.

$$4x-4x^{2}+8x=4$$

Pahekotia te 4x me 8x, ka 12x.

$$12x-4x^{2}=4$$

Tangohia te 4 mai i ngā taha e rua.

$$12x-4x^{2}-4=0$$

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.

$$-4x^{2}+12x-4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -4 mō a, 12 mō b, me -4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

Pūrua 12.

$$x=\frac{-12±\sqrt{144-4\left(-4\right)\left(-4\right)}}{2\left(-4\right)}$$

Whakareatia -4 ki te -4.

$$x=\frac{-12±\sqrt{144+16\left(-4\right)}}{2\left(-4\right)}$$

Whakareatia 16 ki te -4.

$$x=\frac{-12±\sqrt{144-64}}{2\left(-4\right)}$$

Tāpiri 144 ki te -64.

$$x=\frac{-12±\sqrt{80}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te 80.

$$x=\frac{-12±4\sqrt{5}}{2\left(-4\right)}$$

Whakareatia 2 ki te -4.

$$x=\frac{-12±4\sqrt{5}}{-8}$$

Nā, me whakaoti te whārite x=\frac{-12±4\sqrt{5}}{-8} ina he tāpiri te ±. Tāpiri -12 ki te 4\sqrt{5}.

$$x=\frac{4\sqrt{5}-12}{-8}$$

Whakawehe -12+4\sqrt{5} ki te -8.

$$x=\frac{3-\sqrt{5}}{2}$$

Nā, me whakaoti te whārite x=\frac{-12±4\sqrt{5}}{-8} ina he tango te ±. Tango 4\sqrt{5} mai i -12.

$$x=\frac{-4\sqrt{5}-12}{-8}$$

Whakawehe -12-4\sqrt{5} ki te -8.

$$x=\frac{\sqrt{5}+3}{2}$$

Kua oti te whārite te whakatau.

$$x=\frac{3-\sqrt{5}}{2} x=\frac{\sqrt{5}+3}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tangohia te 4x^{2} mai i ngā taha e rua.

$$4x-4x^{2}=-8x+4$$

Me tāpiri te 8x ki ngā taha e rua.

$$4x-4x^{2}+8x=4$$

Pahekotia te 4x me 8x, ka 12x.

$$12x-4x^{2}=4$$

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.

$$-4x^{2}+12x=4$$

Whakawehea ngā taha e rua ki te -4.

$$\frac{-4x^{2}+12x}{-4}=\frac{4}{-4}$$

Mā te whakawehe ki te -4 ka wetekia te whakareanga ki te -4.

$$x^{2}+\frac{12}{-4}x=\frac{4}{-4}$$

Whakawehe 12 ki te -4.

$$x^{2}-3x=\frac{4}{-4}$$

Whakawehe 4 ki te -4.

$$x^{2}-3x=-1$$

Whakawehea te -3, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{3}{2}. Nā, tāpiria te pūrua o te -\frac{3}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-3x+\frac{9}{4}=-1+\frac{9}{4}$$

Tāpiri -1 ki te \frac{9}{4}.

$$x^{2}-3x+\frac{9}{4}=\frac{5}{4}$$

Tauwehea x^{2}-3x+\frac{9}{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}.

$$\left(x-\frac{3}{2}\right)^{2}=\frac{5}{4}$$

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

$$\sqrt{\left(x-\frac{3}{2}\right)^{2}}=\sqrt{\frac{5}{4}}$$

Whakarūnātia.

$$x-\frac{3}{2}=\frac{\sqrt{5}}{2} x-\frac{3}{2}=-\frac{\sqrt{5}}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{5}+3}{2} x=\frac{3-\sqrt{5}}{2}$$