Whakaoti i te 8x^2-14x=6

$$8 x ^ { 2 } - 14 x = 6$$

Whakaoti mō x

$$x = \frac{\sqrt{97} + 7}{8} \approx 2.106107225$$

$$x=\frac{7-\sqrt{97}}{8}\approx -0.356107225$$

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.

$$8x^{2}-14x=6$$

Me tango 6 mai i ngā taha e rua o te whārite.

$$8x^{2}-14x-6=6-6$$

Mā te tango i te 6 i a ia ake anō ka toe ko te 0.

$$8x^{2}-14x-6=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 8 mō a, -14 mō b, me -6 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 8\left(-6\right)}}{2\times 8}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 8\left(-6\right)}}{2\times 8}$$

Whakareatia -4 ki te 8.

$$x=\frac{-\left(-14\right)±\sqrt{196-32\left(-6\right)}}{2\times 8}$$

Whakareatia -32 ki te -6.

$$x=\frac{-\left(-14\right)±\sqrt{196+192}}{2\times 8}$$

Tāpiri 196 ki te 192.

$$x=\frac{-\left(-14\right)±\sqrt{388}}{2\times 8}$$

Tuhia te pūtakerua o te 388.

$$x=\frac{-\left(-14\right)±2\sqrt{97}}{2\times 8}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±2\sqrt{97}}{2\times 8}$$

Whakareatia 2 ki te 8.

$$x=\frac{14±2\sqrt{97}}{16}$$

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{97}}{16} ina he tāpiri te ±. Tāpiri 14 ki te 2\sqrt{97}.

$$x=\frac{2\sqrt{97}+14}{16}$$

Whakawehe 14+2\sqrt{97} ki te 16.

$$x=\frac{\sqrt{97}+7}{8}$$

Nā, me whakaoti te whārite x=\frac{14±2\sqrt{97}}{16} ina he tango te ±. Tango 2\sqrt{97} mai i 14.

$$x=\frac{14-2\sqrt{97}}{16}$$

Whakawehe 14-2\sqrt{97} ki te 16.

$$x=\frac{7-\sqrt{97}}{8}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{97}+7}{8} x=\frac{7-\sqrt{97}}{8}$$

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.

$$8x^{2}-14x=6$$

Whakawehea ngā taha e rua ki te 8.

$$\frac{8x^{2}-14x}{8}=\frac{6}{8}$$

Mā te whakawehe ki te 8 ka wetekia te whakareanga ki te 8.

$$x^{2}+\left(-\frac{14}{8}\right)x=\frac{6}{8}$$

Whakahekea te hautanga \frac{-14}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{7}{4}x=\frac{6}{8}$$

Whakahekea te hautanga \frac{6}{8} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 2.

$$x^{2}-\frac{7}{4}x=\frac{3}{4}$$

Whakawehea te -\frac{7}{4}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{8}. Nā, tāpiria te pūrua o te -\frac{7}{8} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{4}x+\left(-\frac{7}{8}\right)^{2}=\frac{3}{4}+\left(-\frac{7}{8}\right)^{2}$$

Pūruatia -\frac{7}{8} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{3}{4}+\frac{49}{64}$$

Tāpiri \frac{3}{4} ki te \frac{49}{64} 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.

$$x^{2}-\frac{7}{4}x+\frac{49}{64}=\frac{97}{64}$$

Tauwehea x^{2}-\frac{7}{4}x+\frac{49}{64}. 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}.

$$\left(x-\frac{7}{8}\right)^{2}=\frac{97}{64}$$

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

$$\sqrt{\left(x-\frac{7}{8}\right)^{2}}=\sqrt{\frac{97}{64}}$$

Whakarūnātia.

$$x-\frac{7}{8}=\frac{\sqrt{97}}{8} x-\frac{7}{8}=-\frac{\sqrt{97}}{8}$$

Me tāpiri \frac{7}{8} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{97}+7}{8} x=\frac{7-\sqrt{97}}{8}$$