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x^{2}-x-1=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=\frac{-\left(-1\right)±\sqrt{1-4\left(-1\right)}}{2}
Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -1 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-\left(-1\right)±\sqrt{1+4}}{2}
Whakareatia -4 ki te -1.
x=\frac{-\left(-1\right)±\sqrt{5}}{2}
Tāpiri 1 ki te 4.
x=\frac{1±\sqrt{5}}{2}
Ko te tauaro o -1 ko 1.
x=\frac{\sqrt{5}+1}{2}
Nā, me whakaoti te whārite x=\frac{1±\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri 1 ki te \sqrt{5}.
x=\frac{1-\sqrt{5}}{2}
Nā, me whakaoti te whārite x=\frac{1±\sqrt{5}}{2} ina he tango te ±. Tango \sqrt{5} mai i 1.
x=\frac{\sqrt{5}+1}{2} x=\frac{1-\sqrt{5}}{2}
Kua oti te whārite te whakatau.
x^{2}-x-1=0
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}-x-1-\left(-1\right)=-\left(-1\right)
Me tāpiri 1 ki ngā taha e rua o te whārite.
x^{2}-x=-\left(-1\right)
Mā te tango i te -1 i a ia ake anō ka toe ko te 0.
x^{2}-x=1
Tango -1 mai i 0.
x^{2}-x+\left(-\frac{1}{2}\right)^{2}=1+\left(-\frac{1}{2}\right)^{2}
Whakawehea te -1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{1}{2}. Nā, tāpiria te pūrua o te -\frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.
x^{2}-x+\frac{1}{4}=1+\frac{1}{4}
Pūruatia -\frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.
x^{2}-x+\frac{1}{4}=\frac{5}{4}
Tāpiri 1 ki te \frac{1}{4}.
\left(x-\frac{1}{2}\right)^{2}=\frac{5}{4}
Tauwehea te x^{2}-x+\frac{1}{4}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.
\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{5}{4}}
Tuhia te pūtakerua o ngā taha e rua o te whārite.
x-\frac{1}{2}=\frac{\sqrt{5}}{2} x-\frac{1}{2}=-\frac{\sqrt{5}}{2}
Whakarūnātia.
x=\frac{\sqrt{5}+1}{2} x=\frac{1-\sqrt{5}}{2}
Me tāpiri \frac{1}{2} ki ngā taha e rua o te whārite.