Whakaoti i te -3x^2+5x-1=0

$$- 3 x ^ { 2 } + 5 x - 1 = 0$$

Whakaoti mō x

$$x = \frac{\sqrt{13} + 5}{6} \approx 1.434258546$$

$$x=\frac{5-\sqrt{13}}{6}\approx 0.232408121$$

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.

$$-3x^{2}+5x-1=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -3 mō a, 5 mō b, me -1 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Pūrua 5.

$$x=\frac{-5±\sqrt{25-4\left(-3\right)\left(-1\right)}}{2\left(-3\right)}$$

Whakareatia -4 ki te -3.

$$x=\frac{-5±\sqrt{25+12\left(-1\right)}}{2\left(-3\right)}$$

Whakareatia 12 ki te -1.

$$x=\frac{-5±\sqrt{25-12}}{2\left(-3\right)}$$

Tāpiri 25 ki te -12.

$$x=\frac{-5±\sqrt{13}}{2\left(-3\right)}$$

Whakareatia 2 ki te -3.

$$x=\frac{-5±\sqrt{13}}{-6}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{13}}{-6} ina he tāpiri te ±. Tāpiri -5 ki te \sqrt{13}.

$$x=\frac{\sqrt{13}-5}{-6}$$

Whakawehe -5+\sqrt{13} ki te -6.

$$x=\frac{5-\sqrt{13}}{6}$$

Nā, me whakaoti te whārite x=\frac{-5±\sqrt{13}}{-6} ina he tango te ±. Tango \sqrt{13} mai i -5.

$$x=\frac{-\sqrt{13}-5}{-6}$$

Whakawehe -5-\sqrt{13} ki te -6.

$$x=\frac{\sqrt{13}+5}{6}$$

Kua oti te whārite te whakatau.

$$x=\frac{5-\sqrt{13}}{6} x=\frac{\sqrt{13}+5}{6}$$

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.

$$-3x^{2}+5x-1=0$$

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

$$-3x^{2}+5x-1-\left(-1\right)=-\left(-1\right)$$

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

$$-3x^{2}+5x=-\left(-1\right)$$

Tango -1 mai i 0.

$$-3x^{2}+5x=1$$

Whakawehea ngā taha e rua ki te -3.

$$\frac{-3x^{2}+5x}{-3}=\frac{1}{-3}$$

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

$$x^{2}+\frac{5}{-3}x=\frac{1}{-3}$$

Whakawehe 5 ki te -3.

$$x^{2}-\frac{5}{3}x=\frac{1}{-3}$$

Whakawehe 1 ki te -3.

$$x^{2}-\frac{5}{3}x=-\frac{1}{3}$$

Whakawehea te -\frac{5}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{5}{6}. Nā, tāpiria te pūrua o te -\frac{5}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{5}{3}x+\left(-\frac{5}{6}\right)^{2}=-\frac{1}{3}+\left(-\frac{5}{6}\right)^{2}$$

Pūruatia -\frac{5}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=-\frac{1}{3}+\frac{25}{36}$$

Tāpiri -\frac{1}{3} ki te \frac{25}{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.

$$x^{2}-\frac{5}{3}x+\frac{25}{36}=\frac{13}{36}$$

Tauwehea x^{2}-\frac{5}{3}x+\frac{25}{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}.

$$\left(x-\frac{5}{6}\right)^{2}=\frac{13}{36}$$

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

$$\sqrt{\left(x-\frac{5}{6}\right)^{2}}=\sqrt{\frac{13}{36}}$$

Whakarūnātia.

$$x-\frac{5}{6}=\frac{\sqrt{13}}{6} x-\frac{5}{6}=-\frac{\sqrt{13}}{6}$$

Me tāpiri \frac{5}{6} ki ngā taha e rua o te whārite.

$$x=\frac{\sqrt{13}+5}{6} x=\frac{5-\sqrt{13}}{6}$$