Whakaoti i te 5z(z-5)=-30

$$5 z ( z - 5 ) = - 30$$

Whakaoti mō z

$$z=2$$

$$z=3$$

Ngā Upane e Whakamahi ana i te Tikanga Tātai Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 5z ki te z-5.

$$5z^{2}-25z=-30$$

Me tāpiri te 30 ki ngā taha e rua.

$$5z^{2}-25z+30=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 5 mō a, -25 mō b, me 30 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\times 5\times 30}}{2\times 5}$$

Pūrua -25.

$$z=\frac{-\left(-25\right)±\sqrt{625-4\times 5\times 30}}{2\times 5}$$

Whakareatia -4 ki te 5.

$$z=\frac{-\left(-25\right)±\sqrt{625-20\times 30}}{2\times 5}$$

Whakareatia -20 ki te 30.

$$z=\frac{-\left(-25\right)±\sqrt{625-600}}{2\times 5}$$

Tāpiri 625 ki te -600.

$$z=\frac{-\left(-25\right)±\sqrt{25}}{2\times 5}$$

Tuhia te pūtakerua o te 25.

$$z=\frac{-\left(-25\right)±5}{2\times 5}$$

Ko te tauaro o -25 ko 25.

$$z=\frac{25±5}{2\times 5}$$

Whakareatia 2 ki te 5.

$$z=\frac{25±5}{10}$$

Nā, me whakaoti te whārite z=\frac{25±5}{10} ina he tāpiri te ±. Tāpiri 25 ki te 5.

$$z=\frac{30}{10}$$

Whakawehe 30 ki te 10.

$$z=3$$

Nā, me whakaoti te whārite z=\frac{25±5}{10} ina he tango te ±. Tango 5 mai i 25.

$$z=\frac{20}{10}$$

Whakawehe 20 ki te 10.

$$z=2$$

Kua oti te whārite te whakatau.

$$z=3 z=2$$

Ngā Upane mō te Whakaoti i te Pūrua

Whakamahia te āhuatanga tohatoha hei whakarea te 5z ki te z-5.

$$5z^{2}-25z=-30$$

Whakawehea ngā taha e rua ki te 5.

$$\frac{5z^{2}-25z}{5}=-\frac{30}{5}$$

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

$$z^{2}+\left(-\frac{25}{5}\right)z=-\frac{30}{5}$$

Whakawehe -25 ki te 5.

$$z^{2}-5z=-\frac{30}{5}$$

Whakawehe -30 ki te 5.

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

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

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

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

$$z^{2}-5z+\frac{25}{4}=-6+\frac{25}{4}$$

Tāpiri -6 ki te \frac{25}{4}.

$$z^{2}-5z+\frac{25}{4}=\frac{1}{4}$$

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

$$\left(z-\frac{5}{2}\right)^{2}=\frac{1}{4}$$

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

$$\sqrt{\left(z-\frac{5}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Whakarūnātia.

$$z-\frac{5}{2}=\frac{1}{2} z-\frac{5}{2}=-\frac{1}{2}$$

Me tāpiri \frac{5}{2} ki ngā taha e rua o te whārite.

$$z=3 z=2$$