Whakaoti i te 4z^2+60z=800

$$4 z ^ { 2 } + 60 z = 800$$

Whakaoti mō z

$$z = \frac{5 \sqrt{41} - 15}{2} \approx 8.507810594$$

$$z=\frac{-5\sqrt{41}-15}{2}\approx -23.507810594$$

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.

$$4z^{2}+60z=800$$

Me tango 800 mai i ngā taha e rua o te whārite.

$$4z^{2}+60z-800=800-800$$

Mā te tango i te 800 i a ia ake anō ka toe ko te 0.

$$4z^{2}+60z-800=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 60 mō b, me -800 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-60±\sqrt{60^{2}-4\times 4\left(-800\right)}}{2\times 4}$$

Pūrua 60.

$$z=\frac{-60±\sqrt{3600-4\times 4\left(-800\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$z=\frac{-60±\sqrt{3600-16\left(-800\right)}}{2\times 4}$$

Whakareatia -16 ki te -800.

$$z=\frac{-60±\sqrt{3600+12800}}{2\times 4}$$

Tāpiri 3600 ki te 12800.

$$z=\frac{-60±\sqrt{16400}}{2\times 4}$$

Tuhia te pūtakerua o te 16400.

$$z=\frac{-60±20\sqrt{41}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$z=\frac{-60±20\sqrt{41}}{8}$$

Nā, me whakaoti te whārite z=\frac{-60±20\sqrt{41}}{8} ina he tāpiri te ±. Tāpiri -60 ki te 20\sqrt{41}.

$$z=\frac{20\sqrt{41}-60}{8}$$

Whakawehe -60+20\sqrt{41} ki te 8.

$$z=\frac{5\sqrt{41}-15}{2}$$

Nā, me whakaoti te whārite z=\frac{-60±20\sqrt{41}}{8} ina he tango te ±. Tango 20\sqrt{41} mai i -60.

$$z=\frac{-20\sqrt{41}-60}{8}$$

Whakawehe -60-20\sqrt{41} ki te 8.

$$z=\frac{-5\sqrt{41}-15}{2}$$

Kua oti te whārite te whakatau.

$$z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}$$

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.

$$4z^{2}+60z=800$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4z^{2}+60z}{4}=\frac{800}{4}$$

Mā te whakawehe ki te 4 ka wetekia te whakareanga ki te 4.

$$z^{2}+\frac{60}{4}z=\frac{800}{4}$$

Whakawehe 60 ki te 4.

$$z^{2}+15z=\frac{800}{4}$$

Whakawehe 800 ki te 4.

$$z^{2}+15z=200$$

Whakawehea te 15, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{15}{2}. Nā, tāpiria te pūrua o te \frac{15}{2} ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}+15z+\left(\frac{15}{2}\right)^{2}=200+\left(\frac{15}{2}\right)^{2}$$

Pūruatia \frac{15}{2} mā te pūrua i te taurunga me te tauraro o te hautanga.

$$z^{2}+15z+\frac{225}{4}=200+\frac{225}{4}$$

Tāpiri 200 ki te \frac{225}{4}.

$$z^{2}+15z+\frac{225}{4}=\frac{1025}{4}$$

Tauwehea z^{2}+15z+\frac{225}{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}.

$$\left(z+\frac{15}{2}\right)^{2}=\frac{1025}{4}$$

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

$$\sqrt{\left(z+\frac{15}{2}\right)^{2}}=\sqrt{\frac{1025}{4}}$$

Whakarūnātia.

$$z+\frac{15}{2}=\frac{5\sqrt{41}}{2} z+\frac{15}{2}=-\frac{5\sqrt{41}}{2}$$

Me tango \frac{15}{2} mai i ngā taha e rua o te whārite.

$$z=\frac{5\sqrt{41}-15}{2} z=\frac{-5\sqrt{41}-15}{2}$$