Whakaoti i te 14x^2+2x=3

$$14 x ^ { 2 } + 2 x = 3$$

Whakaoti mō x

$$x=\frac{\sqrt{43}-1}{14}\approx 0.396959895$$

$$x=\frac{-\sqrt{43}-1}{14}\approx -0.539817037$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

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.

$$14x^{2}+2x=3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$14x^{2}+2x-3=3-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$14x^{2}+2x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 14 mō a, 2 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\times 14\left(-3\right)}}{2\times 14}$$

Pūrua 2.

$$x=\frac{-2±\sqrt{4-4\times 14\left(-3\right)}}{2\times 14}$$

Whakareatia -4 ki te 14.

$$x=\frac{-2±\sqrt{4-56\left(-3\right)}}{2\times 14}$$

Whakareatia -56 ki te -3.

$$x=\frac{-2±\sqrt{4+168}}{2\times 14}$$

Tāpiri 4 ki te 168.

$$x=\frac{-2±\sqrt{172}}{2\times 14}$$

Tuhia te pūtakerua o te 172.

$$x=\frac{-2±2\sqrt{43}}{2\times 14}$$

Whakareatia 2 ki te 14.

$$x=\frac{-2±2\sqrt{43}}{28}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{43}}{28} ina he tāpiri te ±. Tāpiri -2 ki te 2\sqrt{43}.

$$x=\frac{2\sqrt{43}-2}{28}$$

Whakawehe -2+2\sqrt{43} ki te 28.

$$x=\frac{\sqrt{43}-1}{14}$$

Nā, me whakaoti te whārite x=\frac{-2±2\sqrt{43}}{28} ina he tango te ±. Tango 2\sqrt{43} mai i -2.

$$x=\frac{-2\sqrt{43}-2}{28}$$

Whakawehe -2-2\sqrt{43} ki te 28.

$$x=\frac{-\sqrt{43}-1}{14}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{43}-1}{14} x=\frac{-\sqrt{43}-1}{14}$$

Ngā Upane mō te Whakaoti i te Pūrua

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.

$$14x^{2}+2x=3$$

Whakawehea ngā taha e rua ki te 14.

$$\frac{14x^{2}+2x}{14}=\frac{3}{14}$$

Mā te whakawehe ki te 14 ka wetekia te whakareanga ki te 14.

$$x^{2}+\frac{2}{14}x=\frac{3}{14}$$

Whakahekea te hautanga \frac{2}{14} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}+\frac{1}{7}x=\frac{3}{14}$$

Whakawehea te \frac{1}{7}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{14}. Nā, tāpiria te pūrua o te \frac{1}{14} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{1}{7}x+\left(\frac{1}{14}\right)^{2}=\frac{3}{14}+\left(\frac{1}{14}\right)^{2}$$

Pūruatia \frac{1}{14} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{3}{14}+\frac{1}{196}$$

Tāpiri \frac{3}{14} ki te \frac{1}{196} 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.

$$x^{2}+\frac{1}{7}x+\frac{1}{196}=\frac{43}{196}$$

Tauwehea x^{2}+\frac{1}{7}x+\frac{1}{196}. 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}.

$$\left(x+\frac{1}{14}\right)^{2}=\frac{43}{196}$$

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

$$\sqrt{\left(x+\frac{1}{14}\right)^{2}}=\sqrt{\frac{43}{196}}$$

Whakarūnātia.

$$x+\frac{1}{14}=\frac{\sqrt{43}}{14} x+\frac{1}{14}=-\frac{\sqrt{43}}{14}$$

Me tango \frac{1}{14} mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{43}-1}{14} x=\frac{-\sqrt{43}-1}{14}$$