Whakaoti i te 6500=n[595-15n)

$$6500 = n [ 595 - 15 n )$$

Whakaoti mō n

$$n=\frac{119+\sqrt{1439}i}{6}\approx 19.833333333+6.322358913i$$

$$n=\frac{-\sqrt{1439}i+119}{6}\approx 19.833333333-6.322358913i$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te n ki te 595-15n.

$$6500=595n-15n^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$595n-15n^{2}=6500$$

Tangohia te 6500 mai i ngā taha e rua.

$$595n-15n^{2}-6500=0$$

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.

$$-15n^{2}+595n-6500=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -15 mō a, 595 mō b, me -6500 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-595±\sqrt{595^{2}-4\left(-15\right)\left(-6500\right)}}{2\left(-15\right)}$$

Pūrua 595.

$$n=\frac{-595±\sqrt{354025-4\left(-15\right)\left(-6500\right)}}{2\left(-15\right)}$$

Whakareatia -4 ki te -15.

$$n=\frac{-595±\sqrt{354025+60\left(-6500\right)}}{2\left(-15\right)}$$

Whakareatia 60 ki te -6500.

$$n=\frac{-595±\sqrt{354025-390000}}{2\left(-15\right)}$$

Tāpiri 354025 ki te -390000.

$$n=\frac{-595±\sqrt{-35975}}{2\left(-15\right)}$$

Tuhia te pūtakerua o te -35975.

$$n=\frac{-595±5\sqrt{1439}i}{2\left(-15\right)}$$

Whakareatia 2 ki te -15.

$$n=\frac{-595±5\sqrt{1439}i}{-30}$$

Nā, me whakaoti te whārite n=\frac{-595±5\sqrt{1439}i}{-30} ina he tāpiri te ±. Tāpiri -595 ki te 5i\sqrt{1439}.

$$n=\frac{-595+5\sqrt{1439}i}{-30}$$

Whakawehe -595+5i\sqrt{1439} ki te -30.

$$n=\frac{-\sqrt{1439}i+119}{6}$$

Nā, me whakaoti te whārite n=\frac{-595±5\sqrt{1439}i}{-30} ina he tango te ±. Tango 5i\sqrt{1439} mai i -595.

$$n=\frac{-5\sqrt{1439}i-595}{-30}$$

Whakawehe -595-5i\sqrt{1439} ki te -30.

$$n=\frac{119+\sqrt{1439}i}{6}$$

Kua oti te whārite te whakatau.

$$n=\frac{-\sqrt{1439}i+119}{6} n=\frac{119+\sqrt{1439}i}{6}$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te n ki te 595-15n.

$$6500=595n-15n^{2}$$

Whakawhitihia ngā taha kia puta ki te taha mauī ngā kīanga tau taurangi katoa.

$$595n-15n^{2}=6500$$

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.

$$-15n^{2}+595n=6500$$

Whakawehea ngā taha e rua ki te -15.

$$\frac{-15n^{2}+595n}{-15}=\frac{6500}{-15}$$

Mā te whakawehe ki te -15 ka wetekia te whakareanga ki te -15.

$$n^{2}+\frac{595}{-15}n=\frac{6500}{-15}$$

Whakahekea te hautanga \frac{595}{-15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$n^{2}-\frac{119}{3}n=\frac{6500}{-15}$$

Whakahekea te hautanga \frac{6500}{-15} ki ōna wāhi pāpaku rawa mā te tango me te whakakore i te 5.

$$n^{2}-\frac{119}{3}n=-\frac{1300}{3}$$

Whakawehea te -\frac{119}{3}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te -\frac{119}{6}. Nā, tāpiria te pūrua o te -\frac{119}{6} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}-\frac{119}{3}n+\left(-\frac{119}{6}\right)^{2}=-\frac{1300}{3}+\left(-\frac{119}{6}\right)^{2}$$

Pūruatia -\frac{119}{6} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-\frac{119}{3}n+\frac{14161}{36}=-\frac{1300}{3}+\frac{14161}{36}$$

Tāpiri -\frac{1300}{3} ki te \frac{14161}{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.

$$n^{2}-\frac{119}{3}n+\frac{14161}{36}=-\frac{1439}{36}$$

Tauwehea te n^{2}-\frac{119}{3}n+\frac{14161}{36}. Ko te tikanga, ina ko x^{2}+bx+c he pūrua tika, ka taea te tauwehe i ngā wā katoa hei \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{119}{6}\right)^{2}=-\frac{1439}{36}$$

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

$$\sqrt{\left(n-\frac{119}{6}\right)^{2}}=\sqrt{-\frac{1439}{36}}$$

Whakarūnātia.

$$n-\frac{119}{6}=\frac{\sqrt{1439}i}{6} n-\frac{119}{6}=-\frac{\sqrt{1439}i}{6}$$

Me tāpiri \frac{119}{6} ki ngā taha e rua o te whārite.

$$n=\frac{119+\sqrt{1439}i}{6} n=\frac{-\sqrt{1439}i+119}{6}$$