Whakaoti i te 4z^2+160z=600

$$4 { z }^{ 2 } +160z = 600$$

Whakaoti mō z

$$z=5\sqrt{22}-20\approx 3.452078799$$

$$z=-5\sqrt{22}-20\approx -43.452078799$$

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.

$$4z^{2}+160z=600$$

Me tango 600 mai i ngā taha e rua o te whārite.

$$4z^{2}+160z-600=600-600$$

Mā te tango i te 600 i a ia ake anō ka toe ko te 0.

$$4z^{2}+160z-600=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 4 mō a, 160 mō b, me -600 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$z=\frac{-160±\sqrt{160^{2}-4\times 4\left(-600\right)}}{2\times 4}$$

Pūrua 160.

$$z=\frac{-160±\sqrt{25600-4\times 4\left(-600\right)}}{2\times 4}$$

Whakareatia -4 ki te 4.

$$z=\frac{-160±\sqrt{25600-16\left(-600\right)}}{2\times 4}$$

Whakareatia -16 ki te -600.

$$z=\frac{-160±\sqrt{25600+9600}}{2\times 4}$$

Tāpiri 25600 ki te 9600.

$$z=\frac{-160±\sqrt{35200}}{2\times 4}$$

Tuhia te pūtakerua o te 35200.

$$z=\frac{-160±40\sqrt{22}}{2\times 4}$$

Whakareatia 2 ki te 4.

$$z=\frac{-160±40\sqrt{22}}{8}$$

Nā, me whakaoti te whārite z=\frac{-160±40\sqrt{22}}{8} ina he tāpiri te ±. Tāpiri -160 ki te 40\sqrt{22}.

$$z=\frac{40\sqrt{22}-160}{8}$$

Whakawehe -160+40\sqrt{22} ki te 8.

$$z=5\sqrt{22}-20$$

Nā, me whakaoti te whārite z=\frac{-160±40\sqrt{22}}{8} ina he tango te ±. Tango 40\sqrt{22} mai i -160.

$$z=\frac{-40\sqrt{22}-160}{8}$$

Whakawehe -160-40\sqrt{22} ki te 8.

$$z=-5\sqrt{22}-20$$

Kua oti te whārite te whakatau.

$$z=5\sqrt{22}-20 z=-5\sqrt{22}-20$$

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.

$$4z^{2}+160z=600$$

Whakawehea ngā taha e rua ki te 4.

$$\frac{4z^{2}+160z}{4}=\frac{600}{4}$$

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

$$z^{2}+\frac{160}{4}z=\frac{600}{4}$$

Whakawehe 160 ki te 4.

$$z^{2}+40z=\frac{600}{4}$$

Whakawehe 600 ki te 4.

$$z^{2}+40z=150$$

Whakawehea te 40, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 20. Nā, tāpiria te pūrua o te 20 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$z^{2}+40z+20^{2}=150+20^{2}$$

Pūrua 20.

$$z^{2}+40z+400=150+400$$

Tāpiri 150 ki te 400.

$$z^{2}+40z+400=550$$

Tauwehea z^{2}+40z+400. 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}.

$$\left(z+20\right)^{2}=550$$

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

$$\sqrt{\left(z+20\right)^{2}}=\sqrt{550}$$

Whakarūnātia.

$$z+20=5\sqrt{22} z+20=-5\sqrt{22}$$

Me tango 20 mai i ngā taha e rua o te whārite.

$$z=5\sqrt{22}-20 z=-5\sqrt{22}-20$$