Whakaoti i te 125x^2-78*5x+36125=0

$$125 x ^ { 2 } - 78 \cdot 5 x + 36125 = 0$$

Whakaoti mō x (complex solution)

$$x=\frac{39+4\sqrt{11194}i}{25}\approx 1.56+16.92827221i$$

$$x=\frac{-4\sqrt{11194}i+39}{25}\approx 1.56-16.92827221i$$

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.

$$125x^{2}-390x+36125=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 125 mō a, -390 mō b, me 36125 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-390\right)±\sqrt{\left(-390\right)^{2}-4\times 125\times 36125}}{2\times 125}$$

Pūrua -390.

$$x=\frac{-\left(-390\right)±\sqrt{152100-4\times 125\times 36125}}{2\times 125}$$

Whakareatia -4 ki te 125.

$$x=\frac{-\left(-390\right)±\sqrt{152100-500\times 36125}}{2\times 125}$$

Whakareatia -500 ki te 36125.

$$x=\frac{-\left(-390\right)±\sqrt{152100-18062500}}{2\times 125}$$

Tāpiri 152100 ki te -18062500.

$$x=\frac{-\left(-390\right)±\sqrt{-17910400}}{2\times 125}$$

Tuhia te pūtakerua o te -17910400.

$$x=\frac{-\left(-390\right)±40\sqrt{11194}i}{2\times 125}$$

Ko te tauaro o -390 ko 390.

$$x=\frac{390±40\sqrt{11194}i}{2\times 125}$$

Whakareatia 2 ki te 125.

$$x=\frac{390±40\sqrt{11194}i}{250}$$

Nā, me whakaoti te whārite x=\frac{390±40\sqrt{11194}i}{250} ina he tāpiri te ±. Tāpiri 390 ki te 40i\sqrt{11194}.

$$x=\frac{390+40\sqrt{11194}i}{250}$$

Whakawehe 390+40i\sqrt{11194} ki te 250.

$$x=\frac{39+4\sqrt{11194}i}{25}$$

Nā, me whakaoti te whārite x=\frac{390±40\sqrt{11194}i}{250} ina he tango te ±. Tango 40i\sqrt{11194} mai i 390.

$$x=\frac{-40\sqrt{11194}i+390}{250}$$

Whakawehe 390-40i\sqrt{11194} ki te 250.

$$x=\frac{-4\sqrt{11194}i+39}{25}$$

Kua oti te whārite te whakatau.

$$x=\frac{39+4\sqrt{11194}i}{25} x=\frac{-4\sqrt{11194}i+39}{25}$$

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.

$$125x^{2}-390x+36125=0$$

Me tango 36125 mai i ngā taha e rua o te whārite.

$$125x^{2}-390x+36125-36125=-36125$$

Mā te tango i te 36125 i a ia ake anō ka toe ko te 0.

$$125x^{2}-390x=-36125$$

Whakawehea ngā taha e rua ki te 125.

$$\frac{125x^{2}-390x}{125}=-\frac{36125}{125}$$

Mā te whakawehe ki te 125 ka wetekia te whakareanga ki te 125.

$$x^{2}+\left(-\frac{390}{125}\right)x=-\frac{36125}{125}$$

Whakahekea te hautanga \frac{-390}{125} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$x^{2}-\frac{78}{25}x=-\frac{36125}{125}$$

Whakawehe -36125 ki te 125.

$$x^{2}-\frac{78}{25}x=-289$$

Whakawehea te -\frac{78}{25}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{39}{25}. Nā, tāpiria te pūrua o te -\frac{39}{25} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}-\frac{78}{25}x+\left(-\frac{39}{25}\right)^{2}=-289+\left(-\frac{39}{25}\right)^{2}$$

Pūruatia -\frac{39}{25} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$x^{2}-\frac{78}{25}x+\frac{1521}{625}=-289+\frac{1521}{625}$$

Tāpiri -289 ki te \frac{1521}{625}.

$$x^{2}-\frac{78}{25}x+\frac{1521}{625}=-\frac{179104}{625}$$

Tauwehea x^{2}-\frac{78}{25}x+\frac{1521}{625}. 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}.

$$\left(x-\frac{39}{25}\right)^{2}=-\frac{179104}{625}$$

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

$$\sqrt{\left(x-\frac{39}{25}\right)^{2}}=\sqrt{-\frac{179104}{625}}$$

Whakarūnātia.

$$x-\frac{39}{25}=\frac{4\sqrt{11194}i}{25} x-\frac{39}{25}=-\frac{4\sqrt{11194}i}{25}$$

Me tāpiri \frac{39}{25} ki ngā taha e rua o te whārite.

$$x=\frac{39+4\sqrt{11194}i}{25} x=\frac{-4\sqrt{11194}i+39}{25}$$