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4x^{2}+6x-3=12
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}+6x-3-12=12-12
Me tango 12 mai i ngā taha e rua o te whārite.
4x^{2}+6x-3-12=0
Mā te tango i te 12 i a ia ake anō ka toe ko te 0.
4x^{2}+6x-15=0
Tango 12 mai i -3.
x=\frac{-6±\sqrt{6^{2}-4\times 4\left(-15\right)}}{2\times 4}
Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 6 mō b, me -15 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.
x=\frac{-6±\sqrt{36-4\times 4\left(-15\right)}}{2\times 4}
Pūrua 6.
x=\frac{-6±\sqrt{36-16\left(-15\right)}}{2\times 4}
Whakareatia -4 ki te 4.
x=\frac{-6±\sqrt{36+240}}{2\times 4}
Whakareatia -16 ki te -15.
x=\frac{-6±\sqrt{276}}{2\times 4}
Tāpiri 36 ki te 240.
x=\frac{-6±2\sqrt{69}}{2\times 4}
Tuhia te pūtakerua o te 276.
x=\frac{-6±2\sqrt{69}}{8}
Whakareatia 2 ki te 4.
x=\frac{2\sqrt{69}-6}{8}
Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{69}}{8} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{69}.
x=\frac{\sqrt{69}-3}{4}
Whakawehe -6+2\sqrt{69} ki te 8.
x=\frac{-2\sqrt{69}-6}{8}
Nā, me whakaoti te whārite x=\frac{-6±2\sqrt{69}}{8} ina he tango te ±. Tango 2\sqrt{69} mai i -6.
x=\frac{-\sqrt{69}-3}{4}
Whakawehe -6-2\sqrt{69} ki te 8.
x=\frac{\sqrt{69}-3}{4} x=\frac{-\sqrt{69}-3}{4}
Kua oti te whārite te whakatau.
4x^{2}+6x-3=12
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}+6x-3-\left(-3\right)=12-\left(-3\right)
Me tāpiri 3 ki ngā taha e rua o te whārite.
4x^{2}+6x=12-\left(-3\right)
Mā te tango i te -3 i a ia ake anō ka toe ko te 0.
4x^{2}+6x=15
Tango -3 mai i 12.
\frac{4x^{2}+6x}{4}=\frac{15}{4}
Whakawehea ngā taha e rua ki te 4.
x^{2}+\frac{6}{4}x=\frac{15}{4}
Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.
x^{2}+\frac{3}{2}x=\frac{15}{4}
Whakahekea te hautanga \frac{6}{4} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.
x^{2}+\frac{3}{2}x+\left(\frac{3}{4}\right)^{2}=\frac{15}{4}+\left(\frac{3}{4}\right)^{2}
Whakawehea te \frac{3}{2}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{4}. Nā, tāpiria te pūrua o te \frac{3}{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{3}{2}x+\frac{9}{16}=\frac{15}{4}+\frac{9}{16}
Pūruatia \frac{3}{4} mā te pūrua i te taurunga me te tauraro o te hautanga.
x^{2}+\frac{3}{2}x+\frac{9}{16}=\frac{69}{16}
Tāpiri \frac{15}{4} ki te \frac{9}{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.
\left(x+\frac{3}{4}\right)^{2}=\frac{69}{16}
Tauwehea x^{2}+\frac{3}{2}x+\frac{9}{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}.
\sqrt{\left(x+\frac{3}{4}\right)^{2}}=\sqrt{\frac{69}{16}}
Tuhia te pūtakerua o ngā taha e rua o te whārite.
x+\frac{3}{4}=\frac{\sqrt{69}}{4} x+\frac{3}{4}=-\frac{\sqrt{69}}{4}
Whakarūnātia.
x=\frac{\sqrt{69}-3}{4} x=\frac{-\sqrt{69}-3}{4}
Me tango \frac{3}{4} mai i ngā taha e rua o te whārite.