Whakaoti i te n^2+n-102=0

$$n ^ { 2 } + n - 102 = 0$$

Whakaoti mō n

$$n = \frac{\sqrt{409} - 1}{2} \approx 9.611874208$$

$$n=\frac{-\sqrt{409}-1}{2}\approx -10.611874208$$

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.

$$n^{2}+n-102=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 1 mō b, me -102 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-1±\sqrt{1^{2}-4\left(-102\right)}}{2}$$

Pūrua 1.

$$n=\frac{-1±\sqrt{1-4\left(-102\right)}}{2}$$

Whakareatia -4 ki te -102.

$$n=\frac{-1±\sqrt{1+408}}{2}$$

Tāpiri 1 ki te 408.

$$n=\frac{-1±\sqrt{409}}{2}$$

Nā, me whakaoti te whārite n=\frac{-1±\sqrt{409}}{2} ina he tāpiri te ±. Tāpiri -1 ki te \sqrt{409}.

$$n=\frac{\sqrt{409}-1}{2}$$

Nā, me whakaoti te whārite n=\frac{-1±\sqrt{409}}{2} ina he tango te ±. Tango \sqrt{409} mai i -1.

$$n=\frac{-\sqrt{409}-1}{2}$$

Kua oti te whārite te whakatau.

$$n=\frac{\sqrt{409}-1}{2} n=\frac{-\sqrt{409}-1}{2}$$

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.

$$n^{2}+n-102=0$$

Me tāpiri 102 ki ngā taha e rua o te whārite.

$$n^{2}+n-102-\left(-102\right)=-\left(-102\right)$$

Mā te tango i te -102 i a ia ake anō ka toe ko te 0.

$$n^{2}+n=-\left(-102\right)$$

Tango -102 mai i 0.

$$n^{2}+n=102$$

Whakawehea te 1, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{1}{2}. Nā, tāpiria te pūrua o te \frac{1}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$n^{2}+n+\left(\frac{1}{2}\right)^{2}=102+\left(\frac{1}{2}\right)^{2}$$

Pūruatia \frac{1}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}+n+\frac{1}{4}=102+\frac{1}{4}$$

Tāpiri 102 ki te \frac{1}{4}.

$$n^{2}+n+\frac{1}{4}=\frac{409}{4}$$

Tauwehea n^{2}+n+\frac{1}{4}. 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}.

$$\left(n+\frac{1}{2}\right)^{2}=\frac{409}{4}$$

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

$$\sqrt{\left(n+\frac{1}{2}\right)^{2}}=\sqrt{\frac{409}{4}}$$

Whakarūnātia.

$$n+\frac{1}{2}=\frac{\sqrt{409}}{2} n+\frac{1}{2}=-\frac{\sqrt{409}}{2}$$

Me tango \frac{1}{2} mai i ngā taha e rua o te whārite.

$$n=\frac{\sqrt{409}-1}{2} n=\frac{-\sqrt{409}-1}{2}$$