Whakaoti i te pix^2+3x+0.1415926=0

$$\pi { x }^{ 2 } +3x+0.1415926=0$$

Whakaoti mō x (complex solution)

$$x=\frac{\sqrt{-\frac{707963\pi }{1250000}+9}-3}{2\pi }\approx -0.049793999$$

$$x=-\frac{\sqrt{-\frac{707963\pi }{1250000}+9}+3}{2\pi }\approx -0.905135659$$

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.

$$\pi x^{2}+3x+0.1415926=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \pi mō a, 3 mō b, me 0.1415926 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\pi \times 0.1415926}}{2\pi }$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\pi \times 0.1415926}}{2\pi }$$

Whakareatia -4 ki te \pi .

$$x=\frac{-3±\sqrt{9+\left(-4\pi \right)\times 0.1415926}}{2\pi }$$

Whakareatia -4\pi ki te 0.1415926.

$$x=\frac{-3±\sqrt{9-\frac{707963\pi }{1250000}}}{2\pi }$$

Tāpiri 9 ki te -\frac{707963\pi }{1250000}.

$$x=\frac{-3±\sqrt{-\frac{707963\pi }{1250000}+9}}{2\pi }$$

Tuhia te pūtakerua o te 9-\frac{707963\pi }{1250000}.

$$x=\frac{-3±\frac{\sqrt{56250000-3539815\pi }}{2500}}{2\pi }$$

Nā, me whakaoti te whārite x=\frac{-3±\frac{\sqrt{56250000-3539815\pi }}{2500}}{2\pi } ina he tāpiri te ±. Tāpiri -3 ki te \frac{\sqrt{56250000-3539815\pi }}{2500}.

$$x=\frac{\frac{\sqrt{56250000-3539815\pi }}{2500}-3}{2\pi }$$

Whakawehe -3+\frac{\sqrt{56250000-3539815\pi }}{2500} ki te 2\pi .

$$x=\frac{\sqrt{56250000-3539815\pi }-7500}{5000\pi }$$

Nā, me whakaoti te whārite x=\frac{-3±\frac{\sqrt{56250000-3539815\pi }}{2500}}{2\pi } ina he tango te ±. Tango \frac{\sqrt{56250000-3539815\pi }}{2500} mai i -3.

$$x=\frac{-\frac{\sqrt{56250000-3539815\pi }}{2500}-3}{2\pi }$$

Whakawehe -3-\frac{\sqrt{56250000-3539815\pi }}{2500} ki te 2\pi .

$$x=-\frac{\sqrt{56250000-3539815\pi }+7500}{5000\pi }$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{56250000-3539815\pi }-7500}{5000\pi } x=-\frac{\sqrt{56250000-3539815\pi }+7500}{5000\pi }$$

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.

$$\pi x^{2}+3x+0.1415926=0$$

Me tango 0.1415926 mai i ngā taha e rua o te whārite.

$$\pi x^{2}+3x+0.1415926-0.1415926=-0.1415926$$

Mā te tango i te 0.1415926 i a ia ake anō ka toe ko te 0.

$$\pi x^{2}+3x=-0.1415926$$

Whakawehea ngā taha e rua ki te \pi .

$$\frac{\pi x^{2}+3x}{\pi }=-\frac{0.1415926}{\pi }$$

Mā te whakawehe ki te \pi ka wetekia te whakareanga ki te \pi .

$$x^{2}+\frac{3}{\pi }x=-\frac{0.1415926}{\pi }$$

Whakawehe -0.1415926 ki te \pi .

$$x^{2}+\frac{3}{\pi }x=-\frac{707963}{5000000\pi }$$

Whakawehea te \frac{3}{\pi }, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{2\pi }. Nā, tāpiria te pūrua o te \frac{3}{2\pi } ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{\pi }x+\left(\frac{3}{2\pi }\right)^{2}=-\frac{707963}{5000000\pi }+\left(\frac{3}{2\pi }\right)^{2}$$

Pūrua \frac{3}{2\pi }.

$$x^{2}+\frac{3}{\pi }x+\frac{9}{4\pi ^{2}}=-\frac{707963}{5000000\pi }+\frac{9}{4\pi ^{2}}$$

Tāpiri -\frac{707963}{5000000\pi } ki te \frac{9}{4\pi ^{2}}.

$$x^{2}+\frac{3}{\pi }x+\frac{9}{4\pi ^{2}}=\frac{-\frac{707963\pi }{5000000}+\frac{9}{4}}{\pi ^{2}}$$

Tauwehea x^{2}+\frac{3}{\pi }x+\frac{9}{4\pi ^{2}}. 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}.

$$\left(x+\frac{3}{2\pi }\right)^{2}=\frac{-\frac{707963\pi }{5000000}+\frac{9}{4}}{\pi ^{2}}$$

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

$$\sqrt{\left(x+\frac{3}{2\pi }\right)^{2}}=\sqrt{\frac{-\frac{707963\pi }{5000000}+\frac{9}{4}}{\pi ^{2}}}$$

Whakarūnātia.

$$x+\frac{3}{2\pi }=\frac{\sqrt{56250000-3539815\pi }}{5000\pi } x+\frac{3}{2\pi }=-\frac{\sqrt{56250000-3539815\pi }}{5000\pi }$$

Me tango \frac{3}{2\pi } mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{56250000-3539815\pi }-7500}{5000\pi } x=-\frac{\sqrt{56250000-3539815\pi }+7500}{5000\pi }$$

Whakaoti mō x

$$x=\frac{\sqrt{5\left(11250000-707963\pi \right)}-7500}{5000\pi }\approx -0.049793999$$

$$x=-\frac{\sqrt{5\left(11250000-707963\pi \right)}+7500}{5000\pi }\approx -0.905135659$$

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.

$$\pi x^{2}+3x+0.1415926=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi \pi mō a, 3 mō b, me 0.1415926 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-3±\sqrt{3^{2}-4\pi \times 0.1415926}}{2\pi }$$

Pūrua 3.

$$x=\frac{-3±\sqrt{9-4\pi \times 0.1415926}}{2\pi }$$

Whakareatia -4 ki te \pi .

$$x=\frac{-3±\sqrt{9+\left(-4\pi \right)\times 0.1415926}}{2\pi }$$

Whakareatia -4\pi ki te 0.1415926.

$$x=\frac{-3±\sqrt{9-\frac{707963\pi }{1250000}}}{2\pi }$$

Tāpiri 9 ki te -\frac{707963\pi }{1250000}.

$$x=\frac{-3±\sqrt{-\frac{707963\pi }{1250000}+9}}{2\pi }$$

Tuhia te pūtakerua o te 9-\frac{707963\pi }{1250000}.

$$x=\frac{-3±\frac{\sqrt{56250000-3539815\pi }}{2500}}{2\pi }$$

Nā, me whakaoti te whārite x=\frac{-3±\frac{\sqrt{56250000-3539815\pi }}{2500}}{2\pi } ina he tāpiri te ±. Tāpiri -3 ki te \frac{\sqrt{56250000-3539815\pi }}{2500}.

$$x=\frac{\frac{\sqrt{56250000-3539815\pi }}{2500}-3}{2\pi }$$

Whakawehe -3+\frac{\sqrt{56250000-3539815\pi }}{2500} ki te 2\pi .

$$x=\frac{\sqrt{56250000-3539815\pi }-7500}{5000\pi }$$

Nā, me whakaoti te whārite x=\frac{-3±\frac{\sqrt{56250000-3539815\pi }}{2500}}{2\pi } ina he tango te ±. Tango \frac{\sqrt{56250000-3539815\pi }}{2500} mai i -3.

$$x=\frac{-\frac{\sqrt{56250000-3539815\pi }}{2500}-3}{2\pi }$$

Whakawehe -3-\frac{\sqrt{56250000-3539815\pi }}{2500} ki te 2\pi .

$$x=-\frac{\sqrt{56250000-3539815\pi }+7500}{5000\pi }$$

Kua oti te whārite te whakatau.

$$x=\frac{\sqrt{56250000-3539815\pi }-7500}{5000\pi } x=-\frac{\sqrt{56250000-3539815\pi }+7500}{5000\pi }$$

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.

$$\pi x^{2}+3x+0.1415926=0$$

Me tango 0.1415926 mai i ngā taha e rua o te whārite.

$$\pi x^{2}+3x+0.1415926-0.1415926=-0.1415926$$

Mā te tango i te 0.1415926 i a ia ake anō ka toe ko te 0.

$$\pi x^{2}+3x=-0.1415926$$

Whakawehea ngā taha e rua ki te \pi .

$$\frac{\pi x^{2}+3x}{\pi }=-\frac{0.1415926}{\pi }$$

Mā te whakawehe ki te \pi ka wetekia te whakareanga ki te \pi .

$$x^{2}+\frac{3}{\pi }x=-\frac{0.1415926}{\pi }$$

Whakawehe -0.1415926 ki te \pi .

$$x^{2}+\frac{3}{\pi }x=-\frac{707963}{5000000\pi }$$

Whakawehea te \frac{3}{\pi }, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te \frac{3}{2\pi }. Nā, tāpiria te pūrua o te \frac{3}{2\pi } ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+\frac{3}{\pi }x+\left(\frac{3}{2\pi }\right)^{2}=-\frac{707963}{5000000\pi }+\left(\frac{3}{2\pi }\right)^{2}$$

Pūrua \frac{3}{2\pi }.

$$x^{2}+\frac{3}{\pi }x+\frac{9}{4\pi ^{2}}=-\frac{707963}{5000000\pi }+\frac{9}{4\pi ^{2}}$$

Tāpiri -\frac{707963}{5000000\pi } ki te \frac{9}{4\pi ^{2}}.

$$x^{2}+\frac{3}{\pi }x+\frac{9}{4\pi ^{2}}=\frac{-\frac{707963\pi }{5000000}+\frac{9}{4}}{\pi ^{2}}$$

Tauwehea x^{2}+\frac{3}{\pi }x+\frac{9}{4\pi ^{2}}. 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}.

$$\left(x+\frac{3}{2\pi }\right)^{2}=\frac{-\frac{707963\pi }{5000000}+\frac{9}{4}}{\pi ^{2}}$$

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

$$\sqrt{\left(x+\frac{3}{2\pi }\right)^{2}}=\sqrt{\frac{-\frac{707963\pi }{5000000}+\frac{9}{4}}{\pi ^{2}}}$$

Whakarūnātia.

$$x+\frac{3}{2\pi }=\frac{\sqrt{56250000-3539815\pi }}{5000\pi } x+\frac{3}{2\pi }=-\frac{\sqrt{56250000-3539815\pi }}{5000\pi }$$

Me tango \frac{3}{2\pi } mai i ngā taha e rua o te whārite.

$$x=\frac{\sqrt{56250000-3539815\pi }-7500}{5000\pi } x=-\frac{\sqrt{56250000-3539815\pi }+7500}{5000\pi }$$