Whakaoti i te -25x^2+21x-5=0

$$- 25 x ^ { 2 } + 21 x - 5 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{-\sqrt{59}i+21}{50}\approx 0.42-0.153622915i$$

$$x=\frac{21+\sqrt{59}i}{50}\approx 0.42+0.153622915i$$

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.

$$-25x^{2}+21x-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -25 mō a, 21 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-21±\sqrt{21^{2}-4\left(-25\right)\left(-5\right)}}{2\left(-25\right)}$$

Pūrua 21.

$$x=\frac{-21±\sqrt{441-4\left(-25\right)\left(-5\right)}}{2\left(-25\right)}$$

Whakareatia -4 ki te -25.

$$x=\frac{-21±\sqrt{441+100\left(-5\right)}}{2\left(-25\right)}$$

Whakareatia 100 ki te -5.

$$x=\frac{-21±\sqrt{441-500}}{2\left(-25\right)}$$

Tāpiri 441 ki te -500.

$$x=\frac{-21±\sqrt{-59}}{2\left(-25\right)}$$

Tuhia te pūtakerua o te -59.

$$x=\frac{-21±\sqrt{59}i}{2\left(-25\right)}$$

Whakareatia 2 ki te -25.

$$x=\frac{-21±\sqrt{59}i}{-50}$$

Nā, me whakaoti te whārite x=\frac{-21±\sqrt{59}i}{-50} ina he tāpiri te ±. Tāpiri -21 ki te i\sqrt{59}.

$$x=\frac{-21+\sqrt{59}i}{-50}$$

Whakawehe -21+i\sqrt{59} ki te -50.

$$x=\frac{-\sqrt{59}i+21}{50}$$

Nā, me whakaoti te whārite x=\frac{-21±\sqrt{59}i}{-50} ina he tango te ±. Tango i\sqrt{59} mai i -21.

$$x=\frac{-\sqrt{59}i-21}{-50}$$

Whakawehe -21-i\sqrt{59} ki te -50.

$$x=\frac{21+\sqrt{59}i}{50}$$

Kua oti te whārite te whakatau.

$$x=\frac{-\sqrt{59}i+21}{50} x=\frac{21+\sqrt{59}i}{50}$$

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.

$$-25x^{2}+21x-5=0$$

Me tāpiri 5 ki ngā taha e rua o te whārite.

$$-25x^{2}+21x-5-\left(-5\right)=-\left(-5\right)$$

Mā te tango i te -5 i a ia ake anō ka toe ko te 0.

$$-25x^{2}+21x=-\left(-5\right)$$

Tango -5 mai i 0.

$$-25x^{2}+21x=5$$

Whakawehea ngā taha e rua ki te -25.

$$\frac{-25x^{2}+21x}{-25}=\frac{5}{-25}$$

Mā te whakawehe ki te -25 ka wetekia te whakareanga ki te -25.

$$x^{2}+\frac{21}{-25}x=\frac{5}{-25}$$

Whakawehe 21 ki te -25.

$$x^{2}-\frac{21}{25}x=\frac{5}{-25}$$

Whakahekea te hautanga \frac{5}{-25} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{21}{25}x=-\frac{1}{5}$$

Whakawehea te -\frac{21}{25}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{21}{50}. Nā, tāpiria te pūrua o te -\frac{21}{50} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{21}{25}x+\left(-\frac{21}{50}\right)^{2}=-\frac{1}{5}+\left(-\frac{21}{50}\right)^{2}$$

Pūruatia -\frac{21}{50} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{21}{25}x+\frac{441}{2500}=-\frac{1}{5}+\frac{441}{2500}$$

Tāpiri -\frac{1}{5} ki te \frac{441}{2500} 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.

$$x^{2}-\frac{21}{25}x+\frac{441}{2500}=-\frac{59}{2500}$$

Tauwehea x^{2}-\frac{21}{25}x+\frac{441}{2500}. 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}.

$$\left(x-\frac{21}{50}\right)^{2}=-\frac{59}{2500}$$

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

$$\sqrt{\left(x-\frac{21}{50}\right)^{2}}=\sqrt{-\frac{59}{2500}}$$

Whakarūnātia.

$$x-\frac{21}{50}=\frac{\sqrt{59}i}{50} x-\frac{21}{50}=-\frac{\sqrt{59}i}{50}$$

Me tāpiri \frac{21}{50} ki ngā taha e rua o te whārite.

$$x=\frac{21+\sqrt{59}i}{50} x=\frac{-\sqrt{59}i+21}{50}$$