Whakaoti i te x^2+6x+37=0

$$x ^ { 2 } + 6 x + 37 = 0$$

Whakaoti mō x (complex solution)

$$x=-3+2\sqrt{7}i\approx -3+5.291502622i$$

$$x=-2\sqrt{7}i-3\approx -3-5.291502622i$$

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.

$$x^{2}+6x+37=0$$

Kei te āhua arowhānui tēnei whārite: ax^{2}+bx+c=0. Me whakakapi 1 mō a, 6 mō b, me 37 mō c i te tikanga tātai pūrua, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-6±\sqrt{6^{2}-4\times 37}}{2}$$

Pūrua 6.

$$x=\frac{-6±\sqrt{36-4\times 37}}{2}$$

Whakareatia -4 ki te 37.

$$x=\frac{-6±\sqrt{36-148}}{2}$$

Tāpiri 36 ki te -148.

$$x=\frac{-6±\sqrt{-112}}{2}$$

Tuhia te pūtakerua o te -112.

$$x=\frac{-6±4\sqrt{7}i}{2}$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{7}i}{2} ina he tāpiri te ±. Tāpiri -6 ki te 4i\sqrt{7}.

$$x=\frac{-6+4\sqrt{7}i}{2}$$

Whakawehe -6+4i\sqrt{7} ki te 2.

$$x=-3+2\sqrt{7}i$$

Nā, me whakaoti te whārite x=\frac{-6±4\sqrt{7}i}{2} ina he tango te ±. Tango 4i\sqrt{7} mai i -6.

$$x=\frac{-4\sqrt{7}i-6}{2}$$

Whakawehe -6-4i\sqrt{7} ki te 2.

$$x=-2\sqrt{7}i-3$$

Kua oti te whārite te whakatau.

$$x=-3+2\sqrt{7}i x=-2\sqrt{7}i-3$$

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.

$$x^{2}+6x+37=0$$

Me tango 37 mai i ngā taha e rua o te whārite.

$$x^{2}+6x+37-37=-37$$

Mā te tango i te 37 i a ia ake anō ka toe ko te 0.

$$x^{2}+6x=-37$$

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.

$$x^{2}+6x+3^{2}=-37+3^{2}$$

Pūrua 3.

$$x^{2}+6x+9=-37+9$$

Tāpiri -37 ki te 9.

$$x^{2}+6x+9=-28$$

Tauwehea x^{2}+6x+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}.

$$\left(x+3\right)^{2}=-28$$

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

$$\sqrt{\left(x+3\right)^{2}}=\sqrt{-28}$$

Whakarūnātia.

$$x+3=2\sqrt{7}i x+3=-2\sqrt{7}i$$

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

$$x=-3+2\sqrt{7}i x=-2\sqrt{7}i-3$$