Whakaoti i te 2x+1/x=4x

$$\frac { 2 x + 1 } { x } = 4 x$$

Whakaoti mō x

$$x=\frac{\sqrt{5}+1}{4}\approx 0.809016994$$

$$x=\frac{1-\sqrt{5}}{4}\approx -0.309016994$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$2x+1=4xx$$

Whakareatia te x ki te x, ka x^{2}.

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

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

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

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.

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

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

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

Pūrua 2.

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

Whakareatia -4 ki te -4.

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

Tāpiri 4 ki te 16.

$$x=\frac{-2±\sqrt{20}}{2\left(-4\right)}$$

Tuhia te pūtakerua o te 20.

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

Whakareatia 2 ki te -4.

$$x=\frac{-2±2\sqrt{5}}{-8}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{5}}{-8} ina he tāpiri te ±. Tāpiri -2 ki te 2\sqrt{5}.

$$x=\frac{2\sqrt{5}-2}{-8}$$

Whakawehe -2+2\sqrt{5} ki te -8.

$$x=\frac{1-\sqrt{5}}{4}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{5}}{-8} ina he tango te ±. Tango 2\sqrt{5} mai i -2.

$$x=\frac{-2\sqrt{5}-2}{-8}$$

Whakawehe -2-2\sqrt{5} ki te -8.

$$x=\frac{\sqrt{5}+1}{4}$$

Kua oti te whārite te whakatau.

$$x=\frac{1-\sqrt{5}}{4} x=\frac{\sqrt{5}+1}{4}$$

Ngā Upane mō te Whakaoti i te Pūrua

Tē taea kia ōrite te tāupe x ki 0 nā te kore tautuhi i te whakawehenga mā te kore. Whakareatia ngā taha e rua o te whārite ki te x.

$$2x+1=4xx$$

Whakareatia te x ki te x, ka x^{2}.

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

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

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

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

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

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.

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

Whakawehea ngā taha e rua ki te -4.

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

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

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

Whakahekea te hautanga \frac{2}{-4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

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

Whakawehe -1 ki te -4.

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

Whakawehea te -\frac{1}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{4}. Nā, tāpiria te pūrua o te -\frac{1}{4} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

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

Pūruatia -\frac{1}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{1}{4}+\frac{1}{16}$$

Tāpiri \frac{1}{4} ki te \frac{1}{16} mā te kimi i te tauraro pātahi me te tāpiri i ngā taurunga. Ka whakaiti i te hautanga ki ngā kīanga tau iti rawa e taea ana.

$$x^{2}-\frac{1}{2}x+\frac{1}{16}=\frac{5}{16}$$

Tauwehea x^{2}-\frac{1}{2}x+\frac{1}{16}. 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}.

$$\left(x-\frac{1}{4}\right)^{2}=\frac{5}{16}$$

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

$$\sqrt{\left(x-\frac{1}{4}\right)^{2}}=\sqrt{\frac{5}{16}}$$

Whakarūnātia.

$$x-\frac{1}{4}=\frac{\sqrt{5}}{4} x-\frac{1}{4}=-\frac{\sqrt{5}}{4}$$

Me tāpiri \frac{1}{4} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{5}+1}{4} x=\frac{1-\sqrt{5}}{4}$$