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a+b=4 ab=4\times 1=4
Hei whakaoti i te whārite, whakatauwehea te taha mauī mā te whakarōpū. Tuatahi, me tuhi anō te taha mauī hei 4x^{2}+ax+bx+1. Hei kimi a me b, whakaritea tētahi pūnaha kia whakaoti.
1,4 2,2
I te mea kua tōrunga te ab, he ōrite te tohu o a me b. I te mea kua tōrunga te a+b, he tōrunga hoki a a me b. Whakarārangitia ngā tau tōpū takirua pērā katoa ka hoatu i te hua 4.
1+4=5 2+2=4
Tātaihia te tapeke mō ia takirua.
a=2 b=2
Ko te otinga te takirua ka hoatu i te tapeke 4.
\left(4x^{2}+2x\right)+\left(2x+1\right)
Tuhia anō te 4x^{2}+4x+1 hei \left(4x^{2}+2x\right)+\left(2x+1\right).
2x\left(2x+1\right)+2x+1
Whakatauwehea atu 2x i te 4x^{2}+2x.
\left(2x+1\right)\left(2x+1\right)
Whakatauwehea atu te kīanga pātahi 2x+1 mā te whakamahi i te āhuatanga tātai tohatoha.
\left(2x+1\right)^{2}
Tuhia anōtia hei pūrua huarua.
x=-\frac{1}{2}
Hei kimi i te otinga whārite, whakaotia te 2x+1=0.
4x^{2}+4x+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{-4±\sqrt{4^{2}-4\times 4}}{2\times 4}
Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 4 mō b, me 1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-4±\sqrt{16-4\times 4}}{2\times 4}
Pūrua 4.
x=\frac{-4±\sqrt{16-16}}{2\times 4}
Whakareatia -4 ki te 4.
x=\frac{-4±\sqrt{0}}{2\times 4}
Tāpiri 16 ki te -16.
x=-\frac{4}{2\times 4}
Tuhia te pūtakerua o te 0.
x=-\frac{4}{8}
Whakareatia 2 ki te 4.
x=-\frac{1}{2}
Whakahekea te hautanga \frac{-4}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 4.
4x^{2}+4x+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.
4x^{2}+4x+1-1=-1
Me tango 1 mai i ngā taha e rua o te whārite.
4x^{2}+4x=-1
Mā te tango i te 1 i a ia ake anō ka toe ko te 0.
\frac{4x^{2}+4x}{4}=-\frac{1}{4}
Whakawehea ngā taha e rua ki te 4.
x^{2}+\frac{4}{4}x=-\frac{1}{4}
Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.
x^{2}+x=-\frac{1}{4}
Whakawehe 4 ki te 4.
x^{2}+x+\left(\frac{1}{2}\right)^{2}=-\frac{1}{4}+\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}=\frac{-1+1}{4}
Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.
x^{2}+x+\frac{1}{4}=0
Tāpiri -\frac{1}{4} ki te \frac{1}{4} 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.
\left(x+\frac{1}{2}\right)^{2}=0
Tauwehea x^{2}+x+\frac{1}{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}.
\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{0}
Tuhia te pūtakerua o ngā taha e rua o te whārite.
x+\frac{1}{2}=0 x+\frac{1}{2}=0
Whakarūnātia.
x=-\frac{1}{2} x=-\frac{1}{2}
Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.
x=-\frac{1}{2}
Kua oti te whārite te whakatau. He ōrite ngā whakatau.