Whakaoti i te x^2+8x=3

$$x ^ { 2 } + 8 x = 3$$

Whakaoti mō x (complex solution)

$$x=\sqrt{19}-4\approx 0.358898944$$

$$x=-\left(\sqrt{19}+4\right)\approx -8.358898944$$

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.

$$x^{2}+8x=3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x^{2}+8x-3=3-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$x^{2}+8x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 8 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-3\right)}}{2}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\left(-3\right)}}{2}$$

Whakareatia -4 ki te -3.

$$x=\frac{-8±\sqrt{64+12}}{2}$$

Tāpiri 64 ki te 12.

$$x=\frac{-8±\sqrt{76}}{2}$$

Tuhia te pūtakerua o te 76.

$$x=\frac{-8±2\sqrt{19}}{2}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{19}}{2} ina he tāpiri te ±. Tāpiri -8 ki te 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-8}{2}$$

Whakawehe -8+2\sqrt{19} ki te 2.

$$x=\sqrt{19}-4$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{19}}{2} ina he tango te ±. Tango 2\sqrt{19} mai i -8.

$$x=\frac{-2\sqrt{19}-8}{2}$$

Whakawehe -8-2\sqrt{19} ki te 2.

$$x=-\sqrt{19}-4$$

Kua oti te whārite te whakatau.

$$x=\sqrt{19}-4 x=-\sqrt{19}-4$$

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.

$$x^{2}+8x=3$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+8x+4^{2}=3+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=3+16$$

Tāpiri 3 ki te 16.

$$x^{2}+8x+16=19$$

Tauwehea x^{2}+8x+16. 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}.

$$\left(x+4\right)^{2}=19$$

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

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{19}$$

Whakarūnātia.

$$x+4=\sqrt{19} x+4=-\sqrt{19}$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$x=\sqrt{19}-4 x=-\sqrt{19}-4$$

Whakaoti mō x

$$x=\sqrt{19}-4\approx 0.358898944$$

$$x=-\sqrt{19}-4\approx -8.358898944$$

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.

$$x^{2}+8x=3$$

Me tango 3 mai i ngā taha e rua o te whārite.

$$x^{2}+8x-3=3-3$$

Mā te tango i te 3 i a ia ake anō ka toe ko te 0.

$$x^{2}+8x-3=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 8 mō b, me -3 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\left(-3\right)}}{2}$$

Pūrua 8.

$$x=\frac{-8±\sqrt{64-4\left(-3\right)}}{2}$$

Whakareatia -4 ki te -3.

$$x=\frac{-8±\sqrt{64+12}}{2}$$

Tāpiri 64 ki te 12.

$$x=\frac{-8±\sqrt{76}}{2}$$

Tuhia te pūtakerua o te 76.

$$x=\frac{-8±2\sqrt{19}}{2}$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{19}}{2} ina he tāpiri te ±. Tāpiri -8 ki te 2\sqrt{19}.

$$x=\frac{2\sqrt{19}-8}{2}$$

Whakawehe -8+2\sqrt{19} ki te 2.

$$x=\sqrt{19}-4$$

Nā, me whakaoti te whārite x=\frac{-8±2\sqrt{19}}{2} ina he tango te ±. Tango 2\sqrt{19} mai i -8.

$$x=\frac{-2\sqrt{19}-8}{2}$$

Whakawehe -8-2\sqrt{19} ki te 2.

$$x=-\sqrt{19}-4$$

Kua oti te whārite te whakatau.

$$x=\sqrt{19}-4 x=-\sqrt{19}-4$$

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.

$$x^{2}+8x=3$$

Whakawehea te 8, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 4. Nā, tāpiria te pūrua o te 4 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$x^{2}+8x+4^{2}=3+4^{2}$$

Pūrua 4.

$$x^{2}+8x+16=3+16$$

Tāpiri 3 ki te 16.

$$x^{2}+8x+16=19$$

Tauwehea x^{2}+8x+16. 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}.

$$\left(x+4\right)^{2}=19$$

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

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{19}$$

Whakarūnātia.

$$x+4=\sqrt{19} x+4=-\sqrt{19}$$

Me tango 4 mai i ngā taha e rua o te whārite.

$$x=\sqrt{19}-4 x=-\sqrt{19}-4$$