Whakaoti i te 2n^2-10n-5=-4n

$$2 n ^ { 2 } - 10 n - 5 = - 4 n$$

Whakaoti mō n

$$n = \frac{\sqrt{19} + 3}{2} \approx 3.679449472$$

$$n=\frac{3-\sqrt{19}}{2}\approx -0.679449472$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Me tāpiri te 4n ki ngā taha e rua.

$$2n^{2}-10n-5+4n=0$$

Pahekotia te -10n me 4n, ka -6n.

$$2n^{2}-6n-5=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 2 mō a, -6 mō b, me -5 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 2\left(-5\right)}}{2\times 2}$$

Pūrua -6.

$$n=\frac{-\left(-6\right)±\sqrt{36-4\times 2\left(-5\right)}}{2\times 2}$$

Whakareatia -4 ki te 2.

$$n=\frac{-\left(-6\right)±\sqrt{36-8\left(-5\right)}}{2\times 2}$$

Whakareatia -8 ki te -5.

$$n=\frac{-\left(-6\right)±\sqrt{36+40}}{2\times 2}$$

Tāpiri 36 ki te 40.

$$n=\frac{-\left(-6\right)±\sqrt{76}}{2\times 2}$$

Tuhia te pūtakerua o te 76.

$$n=\frac{-\left(-6\right)±2\sqrt{19}}{2\times 2}$$

Ko te tauaro o -6 ko 6.

$$n=\frac{6±2\sqrt{19}}{2\times 2}$$

Whakareatia 2 ki te 2.

$$n=\frac{6±2\sqrt{19}}{4}$$

Nā, me whakaoti te whārite n=\frac{6±2\sqrt{19}}{4} ina he tāpiri te ±. Tāpiri 6 ki te 2\sqrt{19}.

$$n=\frac{2\sqrt{19}+6}{4}$$

Whakawehe 6+2\sqrt{19} ki te 4.

$$n=\frac{\sqrt{19}+3}{2}$$

Nā, me whakaoti te whārite n=\frac{6±2\sqrt{19}}{4} ina he tango te ±. Tango 2\sqrt{19} mai i 6.

$$n=\frac{6-2\sqrt{19}}{4}$$

Whakawehe 6-2\sqrt{19} ki te 4.

$$n=\frac{3-\sqrt{19}}{2}$$

Kua oti te whārite te whakatau.

$$n=\frac{\sqrt{19}+3}{2} n=\frac{3-\sqrt{19}}{2}$$

Ngā Upane mō te Whakaoti i te Pūrua

Me tāpiri te 4n ki ngā taha e rua.

$$2n^{2}-10n-5+4n=0$$

Pahekotia te -10n me 4n, ka -6n.

$$2n^{2}-6n-5=0$$

Me tāpiri te 5 ki ngā taha e rua. Ko te tau i tāpiria he kore ka hua koia tonu.

$$2n^{2}-6n=5$$

Whakawehea ngā taha e rua ki te 2.

$$\frac{2n^{2}-6n}{2}=\frac{5}{2}$$

Mā te whakawehe ki te 2 ka wetekia te whakareanga ki te 2.

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

Whakawehe -6 ki te 2.

$$n^{2}-3n=\frac{5}{2}$$

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

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

Pūruatia -\frac{3}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$n^{2}-3n+\frac{9}{4}=\frac{5}{2}+\frac{9}{4}$$

Tāpiri \frac{5}{2} ki te \frac{9}{4} 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.

$$n^{2}-3n+\frac{9}{4}=\frac{19}{4}$$

Tauwehea n^{2}-3n+\frac{9}{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}.

$$\left(n-\frac{3}{2}\right)^{2}=\frac{19}{4}$$

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

$$\sqrt{\left(n-\frac{3}{2}\right)^{2}}=\sqrt{\frac{19}{4}}$$

Whakarūnātia.

$$n-\frac{3}{2}=\frac{\sqrt{19}}{2} n-\frac{3}{2}=-\frac{\sqrt{19}}{2}$$

Me tāpiri \frac{3}{2} ki ngā taha e rua o te whārite.

$$n=\frac{\sqrt{19}+3}{2} n=\frac{3-\sqrt{19}}{2}$$