Whakaoti i te 9x^2-14x-14=0

$$9 x ^ { 2 } - 14 x - 14 = 0$$

Whakaoti mō x

$$x = \frac{5 \sqrt{7} + 7}{9} \approx 2.247639617$$

$$x=\frac{7-5\sqrt{7}}{9}\approx -0.692084062$$

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.

$$9x^{2}-14x-14=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 9 mō a, -14 mō b, me -14 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-14\right)±\sqrt{\left(-14\right)^{2}-4\times 9\left(-14\right)}}{2\times 9}$$

Pūrua -14.

$$x=\frac{-\left(-14\right)±\sqrt{196-4\times 9\left(-14\right)}}{2\times 9}$$

Whakareatia -4 ki te 9.

$$x=\frac{-\left(-14\right)±\sqrt{196-36\left(-14\right)}}{2\times 9}$$

Whakareatia -36 ki te -14.

$$x=\frac{-\left(-14\right)±\sqrt{196+504}}{2\times 9}$$

Tāpiri 196 ki te 504.

$$x=\frac{-\left(-14\right)±\sqrt{700}}{2\times 9}$$

Tuhia te pūtakerua o te 700.

$$x=\frac{-\left(-14\right)±10\sqrt{7}}{2\times 9}$$

Ko te tauaro o -14 ko 14.

$$x=\frac{14±10\sqrt{7}}{2\times 9}$$

Whakareatia 2 ki te 9.

$$x=\frac{14±10\sqrt{7}}{18}$$

Nā, me whakaoti te whārite x=\frac{14±10\sqrt{7}}{18} ina he tāpiri te ±. Tāpiri 14 ki te 10\sqrt{7}.

$$x=\frac{10\sqrt{7}+14}{18}$$

Whakawehe 14+10\sqrt{7} ki te 18.

$$x=\frac{5\sqrt{7}+7}{9}$$

Nā, me whakaoti te whārite x=\frac{14±10\sqrt{7}}{18} ina he tango te ±. Tango 10\sqrt{7} mai i 14.

$$x=\frac{14-10\sqrt{7}}{18}$$

Whakawehe 14-10\sqrt{7} ki te 18.

$$x=\frac{7-5\sqrt{7}}{9}$$

Kua oti te whārite te whakatau.

$$x=\frac{5\sqrt{7}+7}{9} x=\frac{7-5\sqrt{7}}{9}$$

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.

$$9x^{2}-14x-14=0$$

Me tāpiri 14 ki ngā taha e rua o te whārite.

$$9x^{2}-14x-14-\left(-14\right)=-\left(-14\right)$$

Mā te tango i te -14 i a ia ake anō ka toe ko te 0.

$$9x^{2}-14x=-\left(-14\right)$$

Tango -14 mai i 0.

$$9x^{2}-14x=14$$

Whakawehea ngā taha e rua ki te 9.

$$\frac{9x^{2}-14x}{9}=\frac{14}{9}$$

Mā te whakawehe ki te 9 ka wetekia te whakareanga ki te 9.

$$x^{2}-\frac{14}{9}x=\frac{14}{9}$$

Whakawehea te -\frac{14}{9}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{9}. Nā, tāpiria te pūrua o te -\frac{7}{9} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{14}{9}x+\left(-\frac{7}{9}\right)^{2}=\frac{14}{9}+\left(-\frac{7}{9}\right)^{2}$$

Pūruatia -\frac{7}{9} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{14}{9}x+\frac{49}{81}=\frac{14}{9}+\frac{49}{81}$$

Tāpiri \frac{14}{9} ki te \frac{49}{81} 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.

$$x^{2}-\frac{14}{9}x+\frac{49}{81}=\frac{175}{81}$$

Tauwehea x^{2}-\frac{14}{9}x+\frac{49}{81}. 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}.

$$\left(x-\frac{7}{9}\right)^{2}=\frac{175}{81}$$

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

$$\sqrt{\left(x-\frac{7}{9}\right)^{2}}=\sqrt{\frac{175}{81}}$$

Whakarūnātia.

$$x-\frac{7}{9}=\frac{5\sqrt{7}}{9} x-\frac{7}{9}=-\frac{5\sqrt{7}}{9}$$

Me tāpiri \frac{7}{9} ki ngā taha e rua o te whārite.

$$x=\frac{5\sqrt{7}+7}{9} x=\frac{7-5\sqrt{7}}{9}$$