Whakaoti i te -5z^2-4z+3=0

$$- 5 z ^ { 2 } - 4 z + 3 = 0$$

Whakaoti mō z

$$z=\frac{\sqrt{19}-2}{5}\approx 0.471779789$$

$$z=\frac{-\sqrt{19}-2}{5}\approx -1.271779789$$

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.

$$-5z^{2}-4z+3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi -5 mō a, -4 mō b, me 3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

Pūrua -4.

$$z=\frac{-\left(-4\right)±\sqrt{16-4\left(-5\right)\times 3}}{2\left(-5\right)}$$

Whakareatia -4 ki te -5.

$$z=\frac{-\left(-4\right)±\sqrt{16+20\times 3}}{2\left(-5\right)}$$

Whakareatia 20 ki te 3.

$$z=\frac{-\left(-4\right)±\sqrt{16+60}}{2\left(-5\right)}$$

Tāpiri 16 ki te 60.

$$z=\frac{-\left(-4\right)±\sqrt{76}}{2\left(-5\right)}$$

Tuhia te pūtakerua o te 76.

$$z=\frac{-\left(-4\right)±2\sqrt{19}}{2\left(-5\right)}$$

Ko te tauaro o -4 ko 4.

$$z=\frac{4±2\sqrt{19}}{2\left(-5\right)}$$

Whakareatia 2 ki te -5.

$$z=\frac{4±2\sqrt{19}}{-10}$$

Nā, me whakaoti te whārite z=\frac{4±2\sqrt{19}}{-10} ina he tāpiri te ±. Tāpiri 4 ki te 2\sqrt{19}.

$$z=\frac{2\sqrt{19}+4}{-10}$$

Whakawehe 4+2\sqrt{19} ki te -10.

$$z=\frac{-\sqrt{19}-2}{5}$$

Nā, me whakaoti te whārite z=\frac{4±2\sqrt{19}}{-10} ina he tango te ±. Tango 2\sqrt{19} mai i 4.

$$z=\frac{4-2\sqrt{19}}{-10}$$

Whakawehe 4-2\sqrt{19} ki te -10.

$$z=\frac{\sqrt{19}-2}{5}$$

Kua oti te whārite te whakatau.

$$z=\frac{-\sqrt{19}-2}{5} z=\frac{\sqrt{19}-2}{5}$$

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.

$$-5z^{2}-4z+3=0$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$-5z^{2}-4z+3-3=-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$-5z^{2}-4z=-3$$

Whakawehea ngā taha e rua ki te -5.

$$\frac{-5z^{2}-4z}{-5}=-\frac{3}{-5}$$

Mā te whakawehe ki te -5 ka wetekia te whakareanga ki te -5.

$$z^{2}+\left(-\frac{4}{-5}\right)z=-\frac{3}{-5}$$

Whakawehe -4 ki te -5.

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

Whakawehe -3 ki te -5.

$$z^{2}+\frac{4}{5}z=\frac{3}{5}$$

Whakawehea te \frac{4}{5}, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{2}{5}. Nā, tāpiria te pūrua o te \frac{2}{5} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}+\frac{4}{5}z+\left(\frac{2}{5}\right)^{2}=\frac{3}{5}+\left(\frac{2}{5}\right)^{2}$$

Pūruatia \frac{2}{5} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$z^{2}+\frac{4}{5}z+\frac{4}{25}=\frac{3}{5}+\frac{4}{25}$$

Tāpiri \frac{3}{5} ki te \frac{4}{25} 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.

$$z^{2}+\frac{4}{5}z+\frac{4}{25}=\frac{19}{25}$$

Tauwehea z^{2}+\frac{4}{5}z+\frac{4}{25}. 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}.

$$\left(z+\frac{2}{5}\right)^{2}=\frac{19}{25}$$

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

$$\sqrt{\left(z+\frac{2}{5}\right)^{2}}=\sqrt{\frac{19}{25}}$$

Whakarūnātia.

$$z+\frac{2}{5}=\frac{\sqrt{19}}{5} z+\frac{2}{5}=-\frac{\sqrt{19}}{5}$$

Me tango \frac{2}{5} mai i ngā taha e rua o te whārite.

$$z=\frac{\sqrt{19}-2}{5} z=\frac{-\sqrt{19}-2}{5}$$