Whakaoti i te z^2-25z+16=0

$${ z }^{ 2 } -25z+16 = 0$$

Whakaoti mō z

$$z = \frac{\sqrt{561} + 25}{2} \approx 24.342719282$$

$$z=\frac{25-\sqrt{561}}{2}\approx 0.657280718$$

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.

$$z^{2}-25z+16=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, -25 mō b, me 16 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 16}}{2}$$

Pūrua -25.

$$z=\frac{-\left(-25\right)±\sqrt{625-4\times 16}}{2}$$

Whakareatia -4 ki te 16.

$$z=\frac{-\left(-25\right)±\sqrt{625-64}}{2}$$

Tāpiri 625 ki te -64.

$$z=\frac{-\left(-25\right)±\sqrt{561}}{2}$$

Ko te tauaro o -25 ko 25.

$$z=\frac{25±\sqrt{561}}{2}$$

Nā, me whakaoti te whārite z=\frac{25±\sqrt{561}}{2} ina he tāpiri te ±. Tāpiri 25 ki te \sqrt{561}.

$$z=\frac{\sqrt{561}+25}{2}$$

Nā, me whakaoti te whārite z=\frac{25±\sqrt{561}}{2} ina he tango te ±. Tango \sqrt{561} mai i 25.

$$z=\frac{25-\sqrt{561}}{2}$$

Kua oti te whārite te whakatau.

$$z=\frac{\sqrt{561}+25}{2} z=\frac{25-\sqrt{561}}{2}$$

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.

$$z^{2}-25z+16=0$$

Me tango 16 mai i ngā taha e rua o te whārite.

$$z^{2}-25z+16-16=-16$$

Mā te tango i te 16 i a ia ake anō ka toe ko te 0.

$$z^{2}-25z=-16$$

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

$$z^{2}-25z+\left(-\frac{25}{2}\right)^{2}=-16+\left(-\frac{25}{2}\right)^{2}$$

Pūruatia -\frac{25}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$z^{2}-25z+\frac{625}{4}=-16+\frac{625}{4}$$

Tāpiri -16 ki te \frac{625}{4}.

$$z^{2}-25z+\frac{625}{4}=\frac{561}{4}$$

Tauwehea z^{2}-25z+\frac{625}{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}.

$$\left(z-\frac{25}{2}\right)^{2}=\frac{561}{4}$$

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

$$\sqrt{\left(z-\frac{25}{2}\right)^{2}}=\sqrt{\frac{561}{4}}$$

Whakarūnātia.

$$z-\frac{25}{2}=\frac{\sqrt{561}}{2} z-\frac{25}{2}=-\frac{\sqrt{561}}{2}$$

Me tāpiri \frac{25}{2} ki ngā taha e rua o te whārite.

$$z=\frac{\sqrt{561}+25}{2} z=\frac{25-\sqrt{561}}{2}$$