Whakaoti i te a^2+6a+4=0

$$a ^ { 2 } + 6 a + 4 = 0$$

Whakaoti mō a (complex solution)

$$a=\sqrt{5}-3\approx -0.763932023$$

$$a=-\left(\sqrt{5}+3\right)\approx -5.236067977$$

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.

$$a^{2}+6a+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-6±\sqrt{6^{2}-4\times 4}}{2}$$

Pūrua 6.

$$a=\frac{-6±\sqrt{36-4\times 4}}{2}$$

Whakareatia -4 ki te 4.

$$a=\frac{-6±\sqrt{36-16}}{2}$$

Tāpiri 36 ki te -16.

$$a=\frac{-6±\sqrt{20}}{2}$$

Tuhia te pūtakerua o te 20.

$$a=\frac{-6±2\sqrt{5}}{2}$$

Nā, me whakaoti te whārite a=\frac{-6±2\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{5}.

$$a=\frac{2\sqrt{5}-6}{2}$$

Whakawehe -6+2\sqrt{5} ki te 2.

$$a=\sqrt{5}-3$$

Nā, me whakaoti te whārite a=\frac{-6±2\sqrt{5}}{2} ina he tango te ±. Tango 2\sqrt{5} mai i -6.

$$a=\frac{-2\sqrt{5}-6}{2}$$

Whakawehe -6-2\sqrt{5} ki te 2.

$$a=-\sqrt{5}-3$$

Kua oti te whārite te whakatau.

$$a=\sqrt{5}-3 a=-\sqrt{5}-3$$

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.

$$a^{2}+6a+4=0$$

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

$$a^{2}+6a+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$a^{2}+6a=-4$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+6a+3^{2}=-4+3^{2}$$

Pūrua 3.

$$a^{2}+6a+9=-4+9$$

Tāpiri -4 ki te 9.

$$a^{2}+6a+9=5$$

Tauwehea a^{2}+6a+9. 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}.

$$\left(a+3\right)^{2}=5$$

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

$$\sqrt{\left(a+3\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$a+3=\sqrt{5} a+3=-\sqrt{5}$$

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

$$a=\sqrt{5}-3 a=-\sqrt{5}-3$$

Whakaoti mō a

$$a=\sqrt{5}-3\approx -0.763932023$$

$$a=-\sqrt{5}-3\approx -5.236067977$$

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.

$$a^{2}+6a+4=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me 4 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-6±\sqrt{6^{2}-4\times 4}}{2}$$

Pūrua 6.

$$a=\frac{-6±\sqrt{36-4\times 4}}{2}$$

Whakareatia -4 ki te 4.

$$a=\frac{-6±\sqrt{36-16}}{2}$$

Tāpiri 36 ki te -16.

$$a=\frac{-6±\sqrt{20}}{2}$$

Tuhia te pūtakerua o te 20.

$$a=\frac{-6±2\sqrt{5}}{2}$$

Nā, me whakaoti te whārite a=\frac{-6±2\sqrt{5}}{2} ina he tāpiri te ±. Tāpiri -6 ki te 2\sqrt{5}.

$$a=\frac{2\sqrt{5}-6}{2}$$

Whakawehe -6+2\sqrt{5} ki te 2.

$$a=\sqrt{5}-3$$

Nā, me whakaoti te whārite a=\frac{-6±2\sqrt{5}}{2} ina he tango te ±. Tango 2\sqrt{5} mai i -6.

$$a=\frac{-2\sqrt{5}-6}{2}$$

Whakawehe -6-2\sqrt{5} ki te 2.

$$a=-\sqrt{5}-3$$

Kua oti te whārite te whakatau.

$$a=\sqrt{5}-3 a=-\sqrt{5}-3$$

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.

$$a^{2}+6a+4=0$$

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

$$a^{2}+6a+4-4=-4$$

Mā te tango i te 4 i a ia ake anō ka toe ko te 0.

$$a^{2}+6a=-4$$

Whakawehea te 6, te tau whakarea o te kīanga tau x, ki te 2 kia riro ai te 3. Nā, tāpiria te pūrua o te 3 ki ngā taha e rua o te whārite. Mā konei e pūrua tika tonu ai te taha mauī o te whārite.

$$a^{2}+6a+3^{2}=-4+3^{2}$$

Pūrua 3.

$$a^{2}+6a+9=-4+9$$

Tāpiri -4 ki te 9.

$$a^{2}+6a+9=5$$

Tauwehea a^{2}+6a+9. 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}.

$$\left(a+3\right)^{2}=5$$

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

$$\sqrt{\left(a+3\right)^{2}}=\sqrt{5}$$

Whakarūnātia.

$$a+3=\sqrt{5} a+3=-\sqrt{5}$$

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

$$a=\sqrt{5}-3 a=-\sqrt{5}-3$$