Whakaoti i te 3x^2-7x-1=0

$$3 { x }^{ 2 } -7x-1=0$$

Whakaoti mō x

$$x = \frac{\sqrt{61} + 7}{6} \approx 2.468374946$$

$$x=\frac{7-\sqrt{61}}{6}\approx -0.135041613$$

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.

$$3x^{2}-7x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 3 mō a, -7 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 3\left(-1\right)}}{2\times 3}$$

Pūrua -7.

$$x=\frac{-\left(-7\right)±\sqrt{49-4\times 3\left(-1\right)}}{2\times 3}$$

Whakareatia -4 ki te 3.

$$x=\frac{-\left(-7\right)±\sqrt{49-12\left(-1\right)}}{2\times 3}$$

Whakareatia -12 ki te -1.

$$x=\frac{-\left(-7\right)±\sqrt{49+12}}{2\times 3}$$

Tāpiri 49 ki te 12.

$$x=\frac{-\left(-7\right)±\sqrt{61}}{2\times 3}$$

Ko te tauaro o -7 ko 7.

$$x=\frac{7±\sqrt{61}}{2\times 3}$$

Whakareatia 2 ki te 3.

$$x=\frac{7±\sqrt{61}}{6}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{61}}{6} ina he tāpiri te ±. Tāpiri 7 ki te \sqrt{61}.

$$x=\frac{\sqrt{61}+7}{6}$$

Nā, me whakaoti te whārite x=\frac{7±\sqrt{61}}{6} ina he tango te ±. Tango \sqrt{61} mai i 7.

$$x=\frac{7-\sqrt{61}}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{61}+7}{6} x=\frac{7-\sqrt{61}}{6}$$

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.

$$3x^{2}-7x-1=0$$

Me tāpiri 1 ki ngā taha e rua o te whārite.

$$3x^{2}-7x-1-\left(-1\right)=-\left(-1\right)$$

Mā te tango i te -1 i a ia ake anō ka toe ko te 0.

$$3x^{2}-7x=-\left(-1\right)$$

Tango -1 mai i 0.

$$3x^{2}-7x=1$$

Whakawehea ngā taha e rua ki te 3.

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

Mā te whakawehe ki te 3 ka wetekia te whakareanga ki te 3.

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

Whakawehea te -\frac{7}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{7}{6}. Nā, tāpiria te pūrua o te -\frac{7}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{7}{3}x+\left(-\frac{7}{6}\right)^{2}=\frac{1}{3}+\left(-\frac{7}{6}\right)^{2}$$

Pūruatia -\frac{7}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{1}{3}+\frac{49}{36}$$

Tāpiri \frac{1}{3} ki te \frac{49}{36} 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.

$$x^{2}-\frac{7}{3}x+\frac{49}{36}=\frac{61}{36}$$

Tauwehea x^{2}-\frac{7}{3}x+\frac{49}{36}. 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}.

$$\left(x-\frac{7}{6}\right)^{2}=\frac{61}{36}$$

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

$$\sqrt{\left(x-\frac{7}{6}\right)^{2}}=\sqrt{\frac{61}{36}}$$

Whakarūnātia.

$$x-\frac{7}{6}=\frac{\sqrt{61}}{6} x-\frac{7}{6}=-\frac{\sqrt{61}}{6}$$

Me tāpiri \frac{7}{6} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{61}+7}{6} x=\frac{7-\sqrt{61}}{6}$$